Projectile Motion Derivation: A Comprehensive Guide

Projectile motion is the motion of an object that is projected into the air and moves under the influence of gravity, without any other external forces acting on it. The motion of a projectile can be broken down into horizontal and vertical components, as gravity only affects the vertical motion. The horizontal motion remains constant, while the vertical motion is constantly accelerated by gravity.

Understanding the Equations of Projectile Motion

The equations of projectile motion are as follows:

1. Horizontal velocity (Vx) = Vx0
2. Horizontal distance (x) = Vx0t
3. Vertical velocity (Vy) = Vy0 – gt
4. Vertical distance (y) = Vy0t – (1/2)gt^2

Where:
– Vx0 = horizontal component of initial velocity
– Vy0 = vertical component of initial velocity
– g = acceleration due to gravity
– t = time

Deriving the Initial Velocity Formula

The initial velocity formula is:

Initial velocity (V) = sqrt(Vx^2 + Vy^2)

Where:
– Vx = horizontal component of initial velocity
– Vy = vertical component of initial velocity

This formula is derived using the Pythagorean theorem, which states that in a right-angled triangle, the square of the length of the hypotenuse (the longest side) is equal to the sum of the squares of the lengths of the other two sides.

Calculating the Maximum Height of a Projectile

The maximum height of a projectile can be found using the following equation:

Maximum height (H) = (Vy0^2) / (2g)

Where:
– Vy0 = vertical component of initial velocity
– g = acceleration due to gravity

This equation is derived from the vertical distance equation, where the vertical velocity (Vy) is set to zero at the maximum height, and the time (t) is the time to reach the maximum height.

Determining the Time of Flight

The time of flight (total time in the air) can be found using the following equation:

Time of flight (T) = (2Vy0) / g

Where:
– Vy0 = vertical component of initial velocity
– g = acceleration due to gravity

This equation is derived from the vertical velocity equation, where the vertical velocity (Vy) is set to zero at the maximum height, and the time (t) is the time of flight.

Calculating the Range of a Projectile

The range (distance traveled) can be found using the following equation:

Range (R) = (Vx0^2) / g

Where:
– Vx0 = horizontal component of initial velocity
– g = acceleration due to gravity

This equation is derived from the horizontal distance equation, where the time (t) is the time of flight.

Determining the Velocity of a Projectile at Any Time

The velocity of a projectile at any time can be found using the following equations:

1. Horizontal velocity (Vx) = Vx0
2. Vertical velocity (Vy) = Vy0 – gt
3. Projectile’s velocity at any time (resultant velocity) = sqrt(Vx^2 + Vy^2)

Where:
– Vx0 = horizontal component of initial velocity
– Vy0 = vertical component of initial velocity
– g = acceleration due to gravity
– t = time

These equations are derived from the horizontal and vertical distance equations, and the Pythagorean theorem.

Angle of Projection for Maximum Range

The angle of projection for maximum range is 45 degrees.

This can be derived by considering the range equation, where the range is maximized when the horizontal and vertical components of the initial velocity are equal (Vx0 = Vy0).

Time to Reach Maximum Height

The time taken to reach maximum height (time to reach the peak) can be found using the following equation:

t_peak = (Vy0) / g

Where:
– Vy0 = vertical component of initial velocity
– g = acceleration due to gravity

This equation is derived from the vertical velocity equation, where the vertical velocity (Vy) is set to zero at the maximum height.

Calculating the Impact Velocity

The impact velocity (final velocity) can be found using the following equation:

Impact velocity (V_impact) = sqrt(Vx^2 + (Vy0 – gt)^2)

Where:
– Vx = horizontal component of initial velocity
– Vy0 = vertical component of initial velocity
– g = acceleration due to gravity
– t = time

This equation is derived from the horizontal and vertical velocity equations, and the Pythagorean theorem.

Time to Reach Peak and Return

The time of flight divided by 2 (time to reach peak and return) can be found using the following equation:

t_half = (Vy0) / g

Where:
– Vy0 = vertical component of initial velocity
– g = acceleration due to gravity

This equation is derived from the time of flight equation, where the time to reach the peak is half the total time of flight.

Example Calculations

Let’s consider an example where a projectile is launched at an initial velocity of 20 m/s at an angle of 30 degrees above the horizontal.

1. Horizontal and vertical components of the initial velocity:
2. Vx0 = V * cos(theta) = 20 m/s * cos(30 degrees) = 17.32 m/s

3. Vy0 = V * sin(theta) = 20 m/s * sin(30 degrees) = 10 m/s

4. Maximum height:

5. H = (Vy0^2) / (2g) = (10 m/s)^2 / (2 * 9.8 m/s^2) = 5.10 m

6. Time of flight:

7. T = (2Vy0) / g = (2 * 10 m/s) / (9.8 m/s^2) = 2.04 s

8. Range:

9. R = (Vx0^2) / g = (17.32 m/s)^2 / (9.8 m/s^2) = 30.14 m

10. Velocity at any time:

11. Vx = Vx0 = 17.32 m/s
12. Vy = Vy0 – gt = 10 m/s – (9.8 m/s^2 * 2.04 s) = -19.98 m/s

13. V = sqrt(Vx^2 + Vy^2) = sqrt((17.32 m/s)^2 + (-19.98 m/s)^2) = 24.52 m/s

14. Time to reach maximum height:

15. t_peak = (Vy0) / g = (10 m/s) / (9.8 m/s^2) = 1.02 s

16. Impact velocity:

17. V_impact = sqrt(Vx^2 + (Vy0 – gt)^2) = sqrt((17.32 m/s)^2 + (-19.98 m/s)^2) = 24.52 m/s

18. Time to reach peak and return:

19. t_half = (Vy0) / g = (10 m/s) / (9.8 m/s^2) = 1.02 s

These calculations demonstrate the application of the derived equations and formulas to determine the various aspects of projectile motion, such as the maximum height, time of flight, range, and velocity at any time.

References:

1. Deriving the 6 Basic Projectile Motion Equations – YouTube, https://www.youtube.com/watch?v=zuFs17Nf41c
2. Derivation of Projectile Motion Equations (class 11 physics) – YouTube, https://www.youtube.com/watch?v=gGScN2-YYIw
3. Projectile Motion | Equations, Initial Velocity & Max Height – Study.com, https://study.com/academy/lesson/parabolic-path-definition-projectiles-quiz.html
4. Projectile Motion – Foundations of Physics – BC Open Textbooks, https://opentextbc.ca/foundationsofphysics/chapter/projectile-motion/