You get on your bike, start pedalling slowly, reaching a certain velocity. Since your velocity changed as a result of you pedalling intensively, surely you would have accelerated in the process. But how to find that acceleration if you have knowledge of time as well? One approach which comes to our mind is applying equations of motion; the other would be to plot a graph with velocity v on the vertical axis and time t on the horizontal axis; such a graph is called velocity-time graph or v-t graph. Similarly, a graph plotted with acceleration a on the vertical axis and time t on the horizontal axis is called acceleration time graph or a-t graph. Before we explore more about these graphs, it is vital to understand two important terms: slope and area. Slope is the tangent of the angle that a straight line or a line drawn to the curve makes with the x-axis; in the following figure for instance, the line drawn to the curve makes an angle of with the +ve x-axis, then
slope
Fig illustrating the slope of v-t graph
Area enclosed by the a-t graph with the t-axis gives the change in velocity.
Area under the a-t graph gives change in velocity
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In the following example, the slope of v-t graph in the interval 3-6 sec can be calculated as follows:
Displacement x = Area under the trapezium
Fig showing a-t graph for a uniformly accelerated motion
For a non uniform motion in which acceleration is not same, the a-t the graph could be as follows
Fig showing a-t graph for a non-uniformly accelerated motion
Question.1 A ball is allowed to fall freely under gravity. Which of the following graphs represents its correct v-t graph?
(a)
(b)
(c)
(d)
Answer: For a body falling freely under gravity, its acceleration is constant and speed keeps on increasing.
Question.2 Which of the following speed time graphs is not possible?
(a)
(b)
(c)
(d) All of these
Answer:
In (a) acceleration is not defined at cusp.
In (b) for the same value of time, different values of velocities are there.
In (c) speed becomes negative.
Hence (d) is the correct answer
Question.3 For the acceleration time graph shown in the figure; what is its corresponding v-t graph? Given that at t=0; v=0.
(a)
(b)
(c)
(d) none of these
Answer: A Between t=0 & t=2;v=0+at;a=3 ms^{-2},t=2v=6 ms^{-1}
Between t=2 & t=4;v=6+(-3)(2)v=0 ms^{-1}
Question.4 The a-t graph of a particle moving along a straight line from rest is shown below. Which graph best represents its variation of velocity with time?
Answer: (a)
The given graph can be written as a=-kt+c;
This implies that the corresponding v-t graph is a parabola.
Question.1 How to find acceleration from v-t graph?
Answer Slope of velocity time (v-t) graph gives acceleration.
Question.2 What is the v-t graph for a ball thrown upwards considering upwards as positive?
Answer: For a ball thrown upwards, acceleration a=-g. Hence slope of v-t graph is -ve.
Question.3 What does the area under the speed time graph give?
Answer: Area under the speed time graph gives distance.
Question.4 What does the velocity time graph look like for a non-uniform accelerated motion?
Answer: For a non-uniform accelerated motion, acceleration is not constant. Therefore the slope of v-t graph is not the same throughout; so a possible graph could be