Keerthi Murthi

How can zero velocity be determined using height and distance?

Exploring Zero Velocity with LambdaGeeks: Zero velocity refers to the state where an object remains at rest, with no change in its position over time. It is possible to determine zero velocity by considering the concepts of height and distance. By analyzing the height and distance covered by an object, we can ascertain whether it is in a state of zero velocity or not. For a detailed understanding of zero velocity and its implications, check out the article on Exploring Zero Velocity with LambdaGeeks.

How to Find Velocity with Height and Distance

velocity is a fundamental concept in physics that describes the rate at which an object changes its position. It is defined as the displacement per unit time. Understanding how to find velocity with height and distance is crucial for gaining insight into the behavior of moving objects. In this article, we will dive into the calculations and equations involved in determining velocity based on height and distance.

Understanding the Concept of Velocity

Before we delve into the specifics of finding velocity with height and distance, let’s first have a clear understanding of what velocity represents. velocity is a vector quantity, meaning it has both magnitude and direction. It can be thought of as the speed of an object in a given direction. For example, if a car is traveling at 60 miles per hour due north, the velocity of the car is 60 mph in the north direction.

The Role of Height and Distance in Determining Velocity

Both height and distance play significant roles in determining the velocity of an object. height refers to the vertical position of an object, while distance refers to the overall length of the path traveled by the object. By incorporating height and distance into the calculation of velocity, we can gain a deeper understanding of an object‘s motion.

The Mathematical Relationship between Velocity, Height, and Distance

To calculate velocity with height and distance, we can utilize the fundamental equation of velocity:

This equation represents the ratio of the distance traveled by an object to the time taken to travel that distance. By rearranging this equation, we can solve for any of the three variables: velocity, distance, or time.

Let’s take a look at an example to illustrate this relationship. Suppose a ball is thrown vertically upwards with an initial velocity of 20 m/s. The ball reaches a maximum height of 30 meters before falling back down. We can use the equation above to calculate the time it takes for the ball to reach its maximum height:

Plugging in the values, we have:

Simplifying the equation gives us:

So, it takes the ball 1.5 seconds to reach its maximum height. Now, if we want to find the velocity of the ball when it reaches the ground, we can use the same equation:

Since the distance is equal to the height of the ball (30 meters) and the time is the total time it takes for the ball to reach its maximum height and fall back down (3 seconds), we have:

Simplifying the equation gives us:

Therefore, the velocity of the ball when it reaches the ground is 10 m/s.

By understanding and applying the mathematical relationship between velocity, height, and distance, we can accurately calculate the velocity of an object in various scenarios. Whether it’s a projectile launched at an angle, an object in free fall, or any other motion, incorporating height and distance into our calculations gives us a comprehensive understanding of the object‘s behavior.

Now that we’ve covered the fundamentals of finding velocity with height and distance, let’s move on to more specific calculations and examples in the following sections.

Also Read:

• How to find final velocity
• How to find final velocity without acceleration
• How to find acceleration with a constant velocity
• How to find tangential velocity
• How to find acceleration with velocity and distance
• How to find horizontal velocity of a projectile
• How to find frequency of transverse wave
• How to find average velocity on a acceleration time graph
• How to find velocity with acceleration and mass
• How to find time with acceleration and distance

When an object is dropped from a certain height, the force of gravity largely influences the object to attain more velocity. So it is clear that height is an entity that influences motion.

A freely falling object initially attains zero velocity, and as it begins to move downward, it gains velocity. Suppose we know the only height of the falling object, how to find velocity with height, and also along with the height, how the other entities influence over velocity are explained in this post.

How to find velocity with Height?

Consider a book kept on a table at the height of h from the ground. When the book falls from the table, then how fast the book falls on the ground is given by velocity. Since the book is at the height of h, how to find the velocity with height?

We know that velocity can be calculated from knowing the distance traveled by the body, and the time taken by it to reach that distance. Mathematically it can be written as,

In the example given above, we are provided with the height h. The height of the body is associated with potential energy. So the basic equation is not valid.

Considering the potential energy possessed by the book before it falls, expression can be written as,

PE = mgh.

But the book is under motion; hence the potential energy is now turned into kinetic energy as

Thus, the potential energy and kinetic energy are equal by the conservation of energy. Hence the equation can be written as

By rearranging the equation, we get velocity as

v² = 2gh

In the above equation, g is the acceleration due to gravity. Any object falling from a certain height is influenced by gravity and is constantly accelerating more due to gravity.

How to find velocity with acceleration and height?

We know how to find velocity with acceleration and distance from the previous article. But we have given with acceleration and height then how to find velocity with acceleration and height instead of distance?

Acceleration and velocity are the proportional entities as the time derivative of velocity is acceleration. If we have acceleration means, on integrating the acceleration, we can have velocity. But in this case, we have acceleration and height. Let us discuss how to find velocity with height if acceleration is given.

Consider a ball is at a certain height above the ground. The ball is dropped from the height ‘h,’ and it begins to accelerate at ‘a’ is in the direction of acceleration due to gravity; this means that the ball is falling from the height h in the direction of gravitational pull.

Since both acceleration and acceleration due to gravity are in the same direction, the total acceleration of the body is equal to the sum of both accelerations of the body and acceleration due to gravity A = g+a. Now the ball’s velocity can be calculated using the equation of motion.

We know from the kinematic equation of motion, distance traveled by the body can be written in terms of the mathematical equation as,

But, we have the height of the ball and the acceleration. The distance can be written in terms of height as,

The ball’s initial position when it begins to move and the final position of the ball gives the distance.

Therefore x = h – 0, i.e., x=h, we can say vertical distance as height. Now substituting the x = h, we have the equation as

Rearranging the above equation, we have

The equation obtained above gives the velocity of the ball given acceleration and height.

Let us set another example if a projectile moving towards the ground from the height h, and its acceleration is more than the acceleration due to gravity because the projectile is overcome from the air friction, then the equation of the velocity will be calculated as,

In the kinematics equations, the velocity is given by

v² = 2Ax

Where x is the distance. But here x = h, then

v² = 2Ah

Consider another case; if you throw a ball in the air, after reaching the height h, the ball begins to accelerate downward due to gravity; the motion is called projectile motion; in this situation, how to find velocity with acceleration and height? The ball’s motion in the air is given in the below figure.

From the above figure, the object’s height is h, and distance is not the height, but we have height in terms of distance by using the equation of projectile motion. The relation between distance and height can be written as,

Substituting the value of distance in the equation of motion, we get

Rearranging the equation, we get velocity as

How to find initial velocity with acceleration and height?

The initial velocity can be derived from the acceleration and height, considering the equation of motion.

A body is accelerating means there must be a change in the velocity of the body with a given instance, which also tells that initially, the body has some velocity that keeps on changing with time. So to find the initial velocity, we need to know the final velocity of the body.

When we throw a ball in the air, it reaches a certain height h with a certain velocity and attains acceleration a. Initially; the ball moves with velocity vi. Finally, the velocity will be vf. The equation of initial velocity will be written using the equation of motion of the ball can be calculated as follow.

The velocity can be

The final velocity of the ball is given as vf, hence from the average velocity.

But at the height h, the ball acquires zero final velocity as it falls back to the ground due to gravity.

But we don’t know the time taken by the ball to reach the height h., so we can use the acceleration. Initially, the ball is accelerating against gravity; its acceleration will become negative.

We know the final velocity is zero, then

Therefore we get the time factor as

Substituting in the equation of average initial velocity, we get

Rearranging the equation, we get

We can calculate the initial velocity when the final velocity is not zero. Consider the equation,

To above equation the substituting the value of t as

t=(v_f+v_i)/a

We get the equation as

(v_f+v_i) (v_f-v_i) = 2ah

The above equation can be written as

v_f²-v_i² = 2ah

Rearranging the terms to get initial velocity as

v_i² = v_f²– 2ah

How to calculate velocity with height and time?

In vertical motion, the distance traveled by the body is equal to the height where the body begins to move.

The velocity can be calculated using height and time. The distance moved by the body with time always describes the body’s velocity. The physical entities such as acceleration and height also contribute to the finding the velocity.

We can calculate the velocity with height and time in three ways

• By vertical motion of the body
• By projectile motion of the body
• By height vs. time graph

By vertical motion of the body

If the basketball is falling from the basket at height h, and is accelerating in the direction of gravity, then the velocity can be given as

But the acceleration is given by

Substituting the value of a and replacing distance term as height h, we get

On rearranging the terms, the velocity with height and time is

By Projectile motion

Consider another example; a basketball player shoots the ball to the basket standing at a distance d away from the basket. The ball makes the projectile motion to reach the basket; then we can calculate the velocity as follows:

The general expression of velocity is given by

The ball travels a distance of d along with the height h; if we neglect the friction, distance can be written as

Substituting the value of x in the general equation of velocity, we get

By height vs. time graph

If we plot a graph with height in the y axis and time in the x-axis, the plot is called a height-time graph.

We can calculate the velocity from the height-time graph. The slope of the height-time graph gives the velocity of the body.

From the above graph, the slope is given by

From the graph, AB is parallel to height h, and BC is parallel to time t; hence we can say that

AB = h and BC = t;

From the definition of velocity, we can say the slope is nothing but velocity. Thus the slope is equal to velocity.

How to find velocity with height and mass?

Though mass does not affect the velocity, it contributes the energy and force required to the body to attain a certain velocity.

The height and the mass are the entities associated with the object’s potential energy. Mass also contributes to the kinetic energy acquired by the object while moving. By knowing mass, let us understand how to find velocity with height.

The object at a certain height possesses potential, which makes the body move, and it is equal to the kinetic energy of the body while moving.

Since both potential energy and kinetic energy are equal, we can equate them.

E_p= E_k

The kinetic energy of the body is

Rearranging the equation, we get

In the beginning, we have told that potential energy = kinetic energy,

Therefore the equation can be rewritten as

Generally, the potential energy is E_p= mgh.

The answer we got from potential energy can be substituted in the above equation to get the velocity of the body.

How to find velocity with height and gravity?

When you throw a stone in the air, it will fall back to the ground due to gravity. It is a general process. But have you observed that the speed of the ball? The speed of the stone while moving down is a little less than the speed of the same stone while it is falling back.

The above statement clarifies that velocity can vary due to gravity also. Gravity comes into action when a body is placed at a certain height; as gravity is an attractive force, it tries to bring the body at height towards the ground—so based on this data, how to find velocity with height and distance?

The earlier section discusses one way of finding the velocity with height and gravity. Let us discuss how to find velocity with height and distance by considering the kinematic equation of motion.

The height is always equal to the distance from the kinematic equation of distance. Hence we can consider the distance as height. So the equation will be

If the motion of the stone is in the direction of gravity, then the acceleration is only due to gravity; hence the equation can be rewritten as

Rearranging the terms, the equation will be

The above equation gives the velocity with height and gravity with the time factor. If the body is accelerating against gravity, then

g = -g

How to find velocity with height and angle?

When a body begins to fall from a certain height towards the surface, it makes some angle θ with the point of dropping. The angle made by the object helps us to find the answer for how to find velocity with height.

The displacement of the body in the vertical position is the height. The vertical component of velocity can be written as

v = v sinθ

If the body is making some horizontal displacement, then velocity is

v = v cosθ

From the equation of motion, the vertical and horizontal velocities can be written as

v_x = v cosθ

v_y = v sinθ-gt; where g is acceleration due to gravity

At maximum height, v_y= 0 = v sinθ –gt

v sinθ = gt

When a body is dropped at an angle θ and travels with velocity v, its range is given by

Therefore, using the value of R,

Therefore, the velocity can be rewritten as

Solved problems on how to calculate velocity with height

Problem 1) A ball is dropped from the height of 15m, and it reaches the ground with a certain velocity. Calculate the velocity of the ball.

Solution:

We are provided with only height h = 15m.

Since the ball moves towards the ground, the motion is due to acceleration due to gravity g. The value of acceleration due to gravity is g = 9.8 m/s². The velocity of the ball is

Substituting the values of h and g;

v = 17.14 m/s.

Problem 2) Calculate the initial velocity of the stone, which is falling from the height of 3m, and its acceleration is 2 m/s², and hence find the time taken by the stone to reach the ground.

Solution:

Given data: Height h = 3m

Acceleration of the stone a = 2 m/s².

The velocity of the stone is given by

v = 3.46 m/s.

The time taken by the stone to reach the ground is given by the equation,

t = 1.79 s.

Problem 3) An object of mass 3 kg is dropped from the height of 7 m, accelerating due to gravity. Calculate the velocity of the object.

Solution:

The data are given –the mass of the object m = 3kg.

Height at which the object has dropped h = 7 m.

Acceleration due to gravity g = 9.8 m/s².

Since the object’s motion is due to mass, height, and gravity, so the work done is equal to potential energy. it is given by

E_p = mgh

The object is moving, so the object possesses kinetic energy; it is represented by the formula,

From the conservation of energy, when an object begins to move, its potential energy is now termed kinetic energy.

Therefore E_p = E_k

The potential energy is E_p = 3×9.8×7

E_p = 205.8 J

Substituting Ep = Ek = 205.8 J.

v² = 137.2

v = 11.71 m/s.

Problem 4) An athlete shoots a shot put in the air in the vertical direction, and it takes a time of 3 seconds to fall on the ground vertically from the height of 7 m from the ground. Calculate the velocity while the shot put is returning to earth.

Solution:

Given data – the height from the ground h = 7 m.

Time is taken to reach the ground = 3 seconds.

The velocity is given by

v = 2.33 m/s.

Problem 5) A body of mass 4 kg is dropped at the height of 11 meters above the ground by making an angle of 20°. Calculate the velocity of the body. (Take acceleration due to gravity as 10 m/s²)

Solution:

The data are given –the mass of the body m = 4 kg.

Height h = 11 m.

Angle θ = 20°.

Acceleration due to gravity g = 10 m/s².

The velocity is given by

v = 43.45 m/s.

What is the formula to calculate velocity with height?

A: The formula to calculate velocity with height is v = √(2gh), where v is the velocity, g is the acceleration due to gravity, and h is the height.

How does calculus relate to finding velocity with height?

Derivatives: From Velocity to Acceleration

Velocity is a measure of how fast an object’s position changes over time. In calculus, we use the derivative to determine this rate of change. The derivative of an object’s position with respect to time gives us its velocity.

where:

• is the velocity as a function of time.
• is the position as a function of time.
• denotes the derivative with respect to time.

Acceleration: The Derivative of Velocity

Gravity affects the motion of objects by accelerating them at a constant rate towards the Earth. This acceleration (denoted as , approximately downward) is the rate of change of velocity. Using calculus, we express this as the derivative of velocity with respect to time.

For an object under only the influence of gravity, the acceleration is constant, so:

Integrals: From Acceleration to Velocity

If we know the acceleration, we can find the velocity by integrating the acceleration function. Since the acceleration due to gravity is constant, the integral of acceleration is a linear function of time:

where is the integration constant, which can be determined if we know the initial velocity of the object.

Velocity and Height: The Integral Relationship

To relate velocity to height, we integrate the velocity function with respect to time, which gives us the position (height in this case) as a function of time.

Substituting the expression for velocity we found by integrating the acceleration, we get:

Here, is another constant of integration, representing the initial height. By solving for these constants using initial conditions, we can fully determine the position function

What is free fall?

A: Free fall is the motion of an object under the influence of gravity alone. In free fall, the only force acting on the object is the force of gravity.

How can I calculate the height from which an object is dropped?

The formula to calculate the height ℎh from which an object is dropped without initial velocity is derived from the kinematic equation:

where:

• ℎh is the height in meters (m),
• g is the acceleration due to gravity (approximately 9.81 /29.81m/s2 on the surface of the Earth),
• t is the time in seconds (s) it takes for the object to hit the ground.

If you have the time it took for the object to fall, you can simply plug the values into this equation to find the height. If you’re measuring the time it takes for an object to hit the ground, you can ignore air resistance for small heights and low speeds.

When calculating the height from which an object is dropped, we assume that it’s in free fall, which means the only force acting on it is gravity. The height can be calculated using the following kinematic equation:

In this formula:

• ℎh represents the height from which the object is dropped (in meters, m).
• g is the acceleration due to gravity, which is approximately near the Earth’s surface.
• t is the time in seconds (s) that it takes for the object to fall to the ground.

To find the height, simply measure the time from the moment the object is dropped until it hits the ground. Then, use that time in the formula above.

For example, if an object takes 3 seconds to hit the ground, the height from which it was dropped is calculated as follows:

So, the object was dropped from a height of approximately 397.305 meters.

How does height affect velocity?

A: The higher the object is, the greater its gravitational potential energy. As the object falls, this potential energy is converted into kinetic energy, leading to an increase in velocity.

What is the final velocity of an object that hits the ground?

A: The final velocity of an object that hits the ground is its velocity at impact. This velocity can be calculated using the formula v = √(2gh), where v is the final velocity, g is the acceleration due to gravity, and h is the height from which the object was dropped.

What role does gravity play in velocity with height?

A: Gravity is the force that pulls objects toward the center of the Earth. In the context of velocity with height, gravity is responsible for the acceleration of the object as it falls and increases its velocity.

How can I calculate the velocity of an object thrown vertically?

A: To calculate the velocity of an object thrown vertically, you can use the equation v = u + gt, where v is the final velocity, u is the initial speed, g is the acceleration due to gravity, and t is the time it takes for the object to reach its peak height.

Also Read:

• How to find velocity with acceleration and time
• How to find angular acceleration from angular velocity
• How to find molar mass from volume
• How to find velocity with constant acceleration
• How to find instantaneous velocity from average velocity
• How to find slope of position time graph
• How to find centripetal acceleration
• How to find horizontal velocity without time
• How to find mass without acceleration
• How to find angular acceleration of a wheel

In the kinematic theory, distance, velocity, acceleration, displacement, and time are the fundamental concepts to derive the equation of motion of in 2-dimensional space.

Generally, the distance traveled by a body per unit time gives the velocity. If the velocity changes with time during the motion, the body possesses the term acceleration. In this post, how velocity, acceleration, and distance are related is discussed in detail, and we get to know how to find velocity with acceleration and distance.

How to find velocity with acceleration and distance?

Suppose the body begins to move with initial velocity zero. The body is moving with acceleration ‘a’ and covers the distance ‘d’ meters; then, we need to find the velocity at which the body is moving. Now arise a question of how to find velocity with acceleration and distance?

Velocity gives how fast an object can move a distance over a given time period.

The expression is given by

v=x/t

But from the considering the equation

v = a*t

t=v/a

Substituting the value of t and rearranging, we get

v=x/(v/a)

v² = a*x

v=√ax

The equation obtained above is applicable if the body begins to move from zero velocity and then accelerates. The body is moving with constant acceleration to reach a distance d.

Using the general expression, we can find the body’s velocity with acceleration and distance with or without time.

How to find velocity from acceleration and distance without time?

The velocity of the body is always measured with the time taken by the body to travel a certain distance. If the time is not given by then, how to find velocity with acceleration and distance?

We follow two methods to find the velocity with given acceleration and distance. Generally, we consider the time in the very first equation; by eliminating the time factor, we get an equation of velocity without time.

By algebraic method:

To calculate the velocity without time, let us consider the equation of velocity with acceleration and time,

v = a * t

The ratio of distance traveled and time gives the velocity of the body. It is given by the equation,

v=x/t

Where x is the distance covered and t is the time taken to cover the distance d,

x/t=at

Substituting the value of v in the first equation; we get,

x = at²

From the kinematic theory, if the velocity of the body is changing with time, then we take the average of the velocity, therefore;

x= at²/2

But we can say that,t= v/a ,substituting in the above equation,

Solving and rearranging the terms we get,

x=v²/2a

v² = 2ax

v=√2ax

The above equation answers how to find velocity with acceleration and distance.

By integral calculus method:

The acceleration can be written as,

a=dv/dt

Velocity is nothing but the time derivative of distance covered by the body; it is given by,

dt=dx/v

Substituting the value of dt in the acceleration equation, we get

a=vdv/dx

a dx = v dv Since we have considered that the initial body possesses zero velocity, we integrate the above equation with the limit zero to a maximum value of the velocity and distance.

ax=v²/2

v² = 2 ax

v=√2ax

How to find velocity from acceleration and distance graph?

The plot of acceleration vs. distance gives the equation of motion under a specific time period.

The area under the acceleration–distance graph gives the square of the velocity of the moving body. From the definition of acceleration, it is the second-order derivative of the distance, so that the velocity will be two times the area.

For example, the acceleration displacement graph for a body moving with constant acceleration, after a certain time, the body decelerates and covers a certain distance, is given below, the velocity of the body can be calculated using the graph.

The area covered by the a-d graph is a triangle; therefore, the area of the triangle is given by

A=1/2 hb

A=1/2 5*7

A = 17. 5 units

The velocity can be written as

A=√2*area

A=√35

Because 2A = 35 units.

v = 5.91 m/s.

How to find initial velocity from acceleration and distance?

Initial velocity is the velocity at which the body begins its motion.

In order to calculate the initial velocity, we have to consider the fundamental equation of the velocity; it is given by;

v=x/t

So the distance is given as; x = v*t

Here, the velocity is not constant; hence we can take the average value of the velocity as

v=v_i+v_f/2

So the equation will be

x=v_i+v_f/2t

But the equation of motion v_f = v_i + at, substituting the value of v_f, we get

x=v_i+(v_i+at)/2t

x=2v_i+at/2t

x=2v_i+a^t/2

2x = 2v_it+at²

On rearranging the above equation,

v_i = x/t – 1/2at

The above equation gives the initial velocity with acceleration and distance.

How to find final velocity from acceleration and distance?

The final velocity is the velocity attained by the body before the motion is stopped due to any hindrance.

When the moving body begins to accelerate means the velocity has been changed. This change in velocity is given by the initial and final velocity of the body. Suppose we have provided only initial velocity, then how to find velocity with acceleration and distance at the final point of the motion is answered below.

To derive the equation for final velocity, let us consider the motion of the car. The car is moving with initial velocity vi, and after some time t, the car begins to accelerate. The car attains the acceleration ‘a’ and covers the distance x.

The derivation can be done by three methods

• By algebraic method
• By calculus method
• By graphical method

Let us study the above three methods in detail.

By algebraic method:

The distance traveled by the body is given by

x=v_i+v_f/2t

The velocity is not constant; it changes with the time period, so choose to take the average of the velocities.

From the kinematic equation of motion, we have

vf = vi + at

Let us rearrange the above equation to get the time as

t = v_f-v_i/2a

Substituting the value in the first equation,

x=v_f-v_i/2 v_f+v_i/a

The above equation is similar to (a+b)(a-b)= a²-b², then the required solution will be

x=v_f-v_i/2a

v_f²– v_i² = 2ax

v_f²= v_i² – 2ax

The equation obtained above is the required equation of final velocity. We can further simplify it by taking the square root on both sides; we get

v_f²=√(v_i²-2ax)

By calculus method:

We know that acceleration is given by the first-order derivative of velocity with respect to time t.

a=dv/dt

And the velocity as

v=dx/dt

Cross multiplying both equations and then integrating by choosing the limit x=0 to x=x and v=v_i to v=v_f we get;

v_f²– v_i² = 2ax

Rearranging the terms;

v_f²= v_i² – 2ax

By graphical method:

A plot of velocity vs. time can helps to find the final velocity of the body.

Generally the distance travelled by the body can be find finding the area covered by the body. Using these available data, we can calculate the distance traveled so that the equation of final velocity can be calculated.

From the above graph, the area of the trapezium OABD gives the distance traveled by the body,

x=OA+BD/2* OD

OA is the initial velocity vi, and BD is the Final velocity vf, and OD is the time, so the equation can be modified as,

x=v_f+v_i/2* t

But, we know that ]t = v_f-v_i/a

x=v_i+v_f/2* v_f-v_i/a

x=v_f²-v_i²/2a

v_f²– v_i² = 2ax

v_f²= v_i² – 2ax

The required equation of final velocity with the graphical method is obtained.

The final velocity equation from acceleration and distance can be rearranged to calculate the initial velocity of the body; it is shown below:

v_i²= v_f² – 2ax

How to find average velocity with acceleration and distance?

If the velocity keeps on changing, then we need to find the average velocity to describe the motion.

In order to establish an equation for average velocity, we must know initial and final velocity. But we can find average velocity even if the initial and final velocity is unknown by knowing acceleration and distance. Let us know how to find average velocity.

Let us suppose that a car is moving with initial velocity v_i and as it begins to accelerate after covering some distance x_i and travels a distance x_f at which it has the final velocity v_f.

The distance covered by the body is from xi to x_f, i.e., at the distance x_i, the velocity of the body is v_i, and at the point x_f, the velocity of the body is v_f, then.

A general expression of average velocity is given as,

v_a=v_i+v_f/2

Equation of motion for final velocity is v_f = v_i+ at

Substituting in the general equation, we have

v_a=v_i+v_i+at/2

v_a=2v_i+at/2

v_a=v_i+1/2 at

By considering the initial velocity expression, we get

v_i = x/t-1/2 at

v_a= x/t-1/2at+1/2 at

 But t=√2x/a

Putting in the above expression, we get

v_a=x/√2x/a

Squaring on both sides, we get

v_a²=x²/2x/a

v_a²=ax²/2x

v_a²=ax/2

v_a=√ax/2

The above equation gives the average velocity of the moving body.

Solved Problems On How to find velocity with acceleration and distance

How to find velocity with acceleration and distance is given, if a motor vehicle is moving with a constant acceleration of 12 m/s² and covers a distance of 87 m, and hence find the time taken by the vehicle to cover the same distance.

Solution:

Given data – The distance covered by the vehicle x = 87 m.

Acceleration of the vehicle a = 12 m/s².

To find the velocity of the motor vehicle,

v=√ax

v=√12*87

v=√1044

v = 32.31 m/s.

From the relation between velocity, acceleration, distance, and time, we have the equation of velocity.

v= x/t

t= x/v

t= 87/32.31

t = 2.69 s.

In a race, the racer rides the bike with an initial velocity of 9 m/s. After time t, the velocity changes, and the acceleration is 3 m/s². The racer covers a distance of 10 m. calculate the final velocity of the bike to reach the given distance and hence find the average velocity of the bike.

Solution:

The equation to find the final velocity of the bike is given by;

v_f²= v_i² – 2ax

v_f²= (9)2 – 2(3 * 10)

v_f²= 81 – 60

v_f²= 21

v_f = 4.58 m/s.

The average velocity is given by

v_a=v_i+v_f/2

v_a=9+4.58/2

v_a=13.58/2

v = 6.79 m/s.

An athlete runs with an initial velocity of 10 m/s. He covers 10 m with a constant acceleration of 4 m/s². Find the initial velocity.

Solution:

Data are given for the calculation – the initial velocity v_i = 10 m/s.

Acceleration a = 4 m/s².

Distance x = 10 m

v_f²= v_i² – 2ax

v_f²= (10)² – 2( 4 *10)

v_f²= 100 – 80

v_f²= 100 – 80

v_f²= 20

v_f = 4.47 m/s.

Calculate the average velocity of particle moving with acceleration of 12 m/s² and the distance travelled by the particle is 26 meters.

Solution:

The formula gives the average velocity for given acceleration and distance.

v_a=√ax/2

The data are given – Acceleration of the particle a = 12 m/s².

Distance traveled by the particle x = 26 m.

Substituting the given values in the equation

√12*26/2

v_a=√156

v_a = 12.48 m/s.

A car travels a distance of 56 meters in 4 seconds. The acceleration of the car with the given time is 2 m/s². Calculate the initial velocity of the car.

Solution:

Given – the distance traveled by the car x = 56 m.

Time is taken by the car to cover the distance x t = 4 s.

Acceleration attain by the car a = 2 m/s².

The initial velocity of the car is given by the formula

v_i = x/t-1/2 at

Substituting the given values in the above equation,

v_i = 56/4-1/2*2*4

v_i = 14 – 4

v_i = 10 m/s.

A graph of acceleration and distance is plotted, then how to find velocity with acceleration and distance is given in the graph.

The distance traveled with acceleration given in the graph forms a trapezium, the area of the trapezium is given by

A=a+b/2* h

Where a and b are the adjacent side of the trapezium and h is the height.

From the above graph

a = 4.5 units

b = 9 units

h = 4 units

Substituting in the given equation,

A=(4.5+9/2)4

A = 27 units.

The velocity is given as

v=√2*area

v=√2*27

v=√56

v = 7.34 m/s.

How do you calculate displacement?

A: Displacement can be calculated using the equation s = ut + 1/2at^2, where s is the displacement, u is the initial velocity, a is the acceleration, and t is the time interval.

How do velocity and displacement relate in kinematics?

A: In kinematics, velocity and displacement are closely related. Velocity is the rate of change of displacement with respect to time. In other words, velocity represents the speed and direction of an object’s motion.

What is kinematics?

A: Kinematics is the branch of physics that deals with the motion of objects without considering the forces that cause the motion. It focuses on describing and analyzing the motion of objects using mathematical equations and concepts.

What happens when an object starts from rest?

A: When an object starts from rest, it means that its initial velocity is zero. In this case, the equation to find velocity simplifies to v = at, where v is the final velocity, a is the acceleration, and t is the time interval.

Also Read:

• How to find constant acceleration with velocity and time
• How to find momentum after collision
• How to find amplitude of transverse wave
• How to find acceleration with velocity and distance
• How to find centripetal acceleration
• How to find acceleration with a constant velocity
• How to find final velocity without acceleration
• How to find constant acceleration with distance and time
• How to find angular acceleration of a wheel
• How to find gravitational acceleration