3 Relative Velocity Graphs: With Explanations You Should Know


Relative velocity describes the velocity of an object with respect to another object which may be under motion or rest.

If you are supposed to interpret relative velocity on the graph, it is called a relative velocity graph. This graph helps to describe the type of motion the object is in at that time. In this post, we will briefly interpret various types of relative velocity graphs.

The relative velocity graph can be classified as a positive, negative, and zero relative velocity graph based on the orientation of the direction of the motion in the pathway.

Positive relative velocity graph

A positive relative velocity means the motion of both the object and the reference object will be in the same direction so that when a graph of such two relative velocities is drawn, the plot will be on a positive coordination axis; such graphical interpretation is called as positive relative velocity.

File:Relative velocity.svg - Wikimedia Commons
Relative velocity of some objects
Image credits: Wikimedia commons

For example, assume that you are supposed to drive a car on a one-way road, and another person is riding a bike on the same road next to you in the same direction; then, you and the bike rider are in relative motion. The velocities of both your car and bike are positively relative to one another. If you measure the velocities of both car and bike and then interpret them on the graph, the resulting plot will be a positive relative velocity graph.

As in a positive relative velocity graph, both objects are in the same direction, and the overall relative velocity between the two objects decreases.

Negative relative velocity graph

When two objects are under motion relative to one another but in the opposite direction, the plot of velocities of those to the equal and opposite motion of such object is called a negative relative velocity graph.

The negative relative velocity is observed on the two-way road, where vehicles are moving in two directions opposite each other. Suppose we measure the velocities by considering two vehicles moving in the opposite direction. In that case, the velocity of one vehicle will be in the opposite direction, like moving towards the negative axis.

The overall relative velocities in the negative relative velocity graph increase as they move in opposite directions.

Non zero relative velocity graph

Two objects moving relative to one another with changing their velocity at a constant rate, the plot of such change in relative velocity is called a non-zero relative velocity graph.

The non-zero relative velocity graph can be obtained when two objects are in a different position at different times. The speed of both objects frequently changes relative to one another. In another sense, we can say that if the angle of both velocities of the objects is different, then the relative velocity between two objects is non-zero.

File:Relative motion.svg - Wikimedia Commons
Positive, negative, non-zero relative velocities
Image credits: Wikimedia commons

Position time graph when relative velocity is zero

When relative velocity is zero, and if we plot it on the position-time graph, we get two straight parallel lines with the same angle of inclination. This means that two objects are moving together with the same velocity at the same time.

When the relative velocity is zero, it does not depends on the direction of the motion of the object. It purely depends on the speed and time interval. The object must travel the same distance at the same speed in the given same interval of time.

The position-time graph when relative velocity is zero is given below.

relative velocity graph
Position-time graph when relative velocity is zero

In the graph, two objects, A and B, are under motion depicted using two straight parallel lines. The inclination of the lines is the same, and their speed changes at a constant rate at the same time interval.

Position time graph when relative velocity is negative

On the position-time graph, the negative relative velocity is represented by two lines in the opposite direction. One is moving along the positive axis, and the other one is moving towards the negative axis representing the opposite direction of the motion.

The graph given below represents the position-time graph when relative velocity is negative.

relative velocity graph
Negative relative velocity graph

From the graph, object A is moving relative to object B. Both objects travel in the opposite direction; hence, the relative velocity between two objects is greater than the magnitude of individual velocities.

Position time graph when relative velocity is non zero

We already know that when relative velocity is non-zero, the velocities of both the moving objects change equally at different positions at a given time interval. On the position-time graph, we get two parallel straight lines at an unequal interval of time, and their inclination is also unequal.

The position-time graph when relative velocity is non-zero is given below.

relative velocity graph
Non-zero Position-time relative velocity graph

The graph clearly shows that two objects are moving relative to one another. The velocity is not zero or constant, but it is changing at a constant rate. Object B is changing its velocity more frequently than object A., so we get two unequal parallel lines.

How to find relative velocity on a graph?

To find relative velocity on the graph, we just need to plot the position-time graph. On the x-t graph, the slope gives the velocity. The difference between the slopes of the two lines depicted on the x-t graph representing the relative motion gives the relative velocity.

Consider the position-time graph of two objects under motion. Let object A has the slope of m1, and object B has the slope of m2. The relative velocity is calculated as follows.

The slope of the object A is

[latex]m_1=\frac{PQ}{QR}[/latex]

The slope of the object B is

[latex]m_2=\frac{XY}{YZ}[/latex]

The relative velocity of A with respect to B is

vrel(AB)=m1-m2

And the relative velocity of B with respect to A is

vrel(BA)=m2-m1

Solved problems on relative velocity graph

Problem 1) The position-time graph of the two bodies is given below. Find the relative velocity of the second body with respect to the first body.

relative velocity graph
Position-time graph for given problem

Solution:

From the above graph, the position and time of the two objects, the slope can be calculated as

[latex]m_1=\frac{QR}{PQ}[/latex]

[latex]m_1=\frac{1}{2}[/latex]

m1=0.5 units

[latex]m_2=\frac{YZ}{XY}[/latex]

[latex]m_2=\frac{2}{2}[/latex]

m2=1 unit.

The relative velocity of object

vrel(BA)=m2-m1

vrel(BA)=1-0.5

vrel(BA)=0.5 m/s.

Problem 2) Find the relative velocity of given objects represented on the position-time graph given below.

relative velocity graph
Position-time graph

Solution:

The slope of the first object is calculated as

[latex]m_1=\frac{QR}{PQ}[/latex]

[latex]m_1=\frac{1.5}{2}[/latex]

m1=0.75 units.

The slope of the second object is given as

[latex]m_2=\frac{YZ}{XY}[/latex]

[latex]m_2=\frac{1.9}{2.1}[/latex]

m2=0.904 units.

Since the motion of object B is opposite to the motion of A, hence the value of the slope of B should be negative with respect to A. Thus, slope m2 can be rewritten as

m2=-0.904 units.

The relative velocity is thus calculated as

vrel= m1-m2= 0.75-(-0.904)

vrel=0.75+0.904

vrel=1.654 m/s.

Problem 3) Find the relative velocity from the given position-time graph below.

relative velocity graph
Zero relative velocity x-t graph

Solution:

From the above graph, it seems both the objects are moving at the same speed simultaneously. In that case, the relative velocity will be zero.

i.e., vA=vB

vrel=0.

Problem 4) Calculate the relative velocity from the graph.

relative velocity graph
x-t graph to find relative velocity

Solution:

From the above graph the slope for first body is

[latex]m_1=\frac{PQ}{QR}[/latex]

[latex]m_1=\frac{2}{1.5}[/latex]

m1=1.33 units.

[latex]m_2=\frac{XY}{YZ}[/latex]

[latex]m_2=\frac{0.5}{1.6}[/latex]

m2=0.312 units.

The relative velocity of the two objects, A and B, is

vAB= 1.33-0.312

vAB = 1.018 units.

Conclusion

In this post we learnt plotting the relative velocity graph of different types which highly depends on the direction of the motion. And also a brief explanation on plotting of position-time graph which defines behavior of all the types of relative velocity on the graph.

Keerthi Murthi

I am Keerthi K Murthy, I have completed post graduation in Physics, with the specialization in the field of solid state physics. I have always consider physics as a fundamental subject which is connected to our daily life. Being a science student I enjoy exploring new things in physics. As a writer my goal is to reach the readers with the simplified manner through my articles. Reach me – keerthikmurthy24@gmail.com

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