The direction of the velocity is important, even in the case for one-dimensional motion. Galileo investigated the issue of relative velocities.
The question of how velocities are all relative was wonderfully demonstrated in a video from the 1950’s by Dr. Hume for PSSC Physics called Frames of Reference.
Think of what happens when you step on a moving sidewalk at the airport. In the frame of reference of the moving sidewalk, you are moving at a casual pace. When you walk with the sidewalk, your velocity in the frame of reference of the ground is actually faster. If you were to foolishly walk against the sidewalk, your velocity in the ground’s frame of reference would actually be slow.
Another example If two cars are driving in the same direction, their relative velocities are the difference of the magnitude of their speeds. Consider when you are driving on the highway. You might be driving at 50 mph, when you are passed by a car driving at 55 mph. However it seems that they are passing you very slowly. Contra-wise, If two cars are driving in opposite directions from each other, their relative velocities are the sum of the magnitude of their velocities.
Another interesting problem you will explore in the homework has to do with finding average speeds. When you first learned about averages, you probably added two numbers together and divided the result by two. However, if you are working with a weighted average then you cannot use this simple trick by plugging in numbers.
Constant velocity: Position vs Time graph:
If we make a graph of position vs time and our object is moving at a constant velocity, the graph will form a straight line. We generally put position on the y-axis, and time on the x-axis. We call this a linear graph. The slope of this line will be the average velocity of our object. If the graph is flat or horizontal, then the object is not moving, the slope is zero, and the velocity is zero. It is important that when you have a graph, there is a title, axes labels, and units.
The faster the object goes, the greater the slope of the graph. In the following graph, you can see an object which is incrementally speeding up. Slow, fast, and faster.
Best fit line:
This is particularly useful for when there is a lot of noise in our data. In middle school, you may have learned to draw a best fit line by hand using a ruler. Computers calculate the best fit line using an algorithm called the Least Squares Fit. The computer calculates the error at every point, and tries to minimize the square of that error.
We can calculate the slope of the following graph. From middle school math, you may remember that you find the slope by taking the rise over the run.
Whenever you have a graph, you can always analyze the graph. It is not enough to merely say the graph goes up! The slope of the graph provides you with information, such as the velocity.
For a position vs. time graph, the
slope = rise/run = Δx/Δt
which of course we know as velocity! We should point out that if the slope is positive, then the velocity is positive. If the slope is negative and the graph goes down, then the velocity is negative relative to a reference point.
This could be particularly useful if the velocity of the graph is not constant. By finding the slope of a line tangent to the graph, we can actually find the instantaneous velocity at any given point in time. We will explore this in depth when we introduce acceleration.
Constant velocity: Velocity vs Time graph:
If the velocity of an object is constant, our v vs t graph is rather simple. It would look like this.
Note the units for both velocity and time. In this case, the slope of our graph is zero. What this means is the velocity is not changing. It is not accelerating.
But there is still a lot of information we can glean from this graph. In this case we will examine the “area under the curve.” The shape of the area under this graph is a simple rectangle.
The area of this rectangle is easy to calculate. In this case, the area is the base x height of the graph.
Area = velocity x time
If we review our kinematic equation, we quickly realize that the area under the curve is actually the displacement.
This general rule that the area under the curve of a velocity vs time graph is the displacement holds even when the velocity is not constant. In those cases, the area is not a simple rectangle and can be difficult to calculate exactly. But often, one can approximate with great accuracy this area for a non-constant velocity.
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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.
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.
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.
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.
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.
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
The slope of the object B is
m1=PQQRm1=PQQR
m2=XYYZ
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.
Solution:
From the above graph, the position and time of the two objects, the slope can be calculated as
m1=QRPQ
m1=12
m1=0.5 units
m2=YZXY
m2=22
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.
Solution:
The slope of the first object is calculated as
m1=QRPQm1=1.52
m1=0.75 units.
The slope of the second object is given as
m2=YZXY
m2=1.92.1
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.
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.
Solution:
From the above graph the slope for first body is
m1=PQQR
m1=21.5
m1=1.33 units.
m2=XYYZ
m2=0.51.6
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.