What is the nature of the velocity-time graph for an object moving with non uniform acceleration?

Distance-time graph is the plot of distance travelled by a body against time. So it will tell us about the journey made by a body and its speed. Below is an example of a distance-time graph, if it is a straight horizontal line then the body is stationary, and its speed is zero, if the line is diagonal then it is moving with a constant speed and if it is anything other than a straight line then the speed is varying.

Distance-Time Graph for Uniform Motion

Let us now consider a distance-time graph in which the body is moving with uniform motion. A body is said to be in uniform motion when the body covers an equal distance in equal time intervals. Let’s consider a time interval of 1 second, If a body covers 10 meters in the first 1-second then it should cover 10 meters in every second from there on, this will indicate that the body is in uniform motion. Let’s draw a graph for uniform motion.

As in uniform motion, the distance-time graph would be a straight line, because the equal distance is covered in equal units of time.

You can see that there are three bodies A, B and C, all of them are in uniform motion then why do they have different slopes?

It is because the slope of a distance-time graph determines the speed of that body, so the steeper the slope greater will be the speed of the body. From the above graph, we can come to a conclusion that body A has the highest speed and body C has the least speed.

Velocity-Time Graph

It is the graph of velocity against time; it shows us how the velocity changes with respect to time. The slope of a velocity-time graph determines its acceleration.

Above is the velocity-time graph for three objects A, B, and C. A flat horizontal line in a velocity-time graph states that the body is moving at a constant velocity. If the straight line has a slope, then that indicates the body is changing its velocity at a constant rate, or it means that the body has constant acceleration.

Distance-Time Graph for Uniformly Accelerated Motion

The distance-time graph for a uniformly accelerated motion looks as shown in the graph above, consider how the distance is changing exponentially indicating that the velocity is changing at a constant rate or there is constant acceleration. But in the velocity-time graph as velocity changes at a constant rate with respect to time in uniformly accelerating motion, the graph would be a straight line with its slope indicating the amount of acceleration.

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Hint: Use relation between acceleration, velocity and time to find the nature of the graph.The Velocity-time graph gives acceleration as slope. In non-uniform motion acceleration changes, eventually slope changes. The slope is either increasing or decreasing.

Formula used:

$a\quad =\dfrac {v}{ t }$

Complete answer:

We know formula for acceleration is given by,$a\quad =\dfrac {v}{ t }$where, a: Acceleration v: Velocity t: TimeFrom the above equation, it can be understood that the slope calculated from the velocity-time graph gives acceleration. It is given that the body is in non-uniform motion which implies acceleration of the body changes at every instant. This means there is a variation in slope too. Slope remains constant for a straight line which leads to the conclusion that a curved line has varying slope i.e. the slope is either increasing or decreasing. Therefore, the velocity-time graph for a body in non-uniform motion is a curved line.

So, the correct answer is “Option D”.

Additional Information:

The figures given below show a velocity-time graph for a body in motion.

Figure.(a) represents Uniform motion or Zero Acceleration. It means the body moves with constant velocity in the same line. In this case, the acceleration is zero.Figure.(b) represents Uniform Acceleration. It means the body travels equal with equal velocity in an equal interval of time.Figure.(c) represents Non-uniform Acceleration. It means the body travels unequal distances in equal intervals of time causing inconsistent velocity.

The above figures show how there is variation in position vs time graph and acceleration vs time graph as there is change in velocity vs time graph.

Note:

As the body is in non-uniform motion, the nature of the graph is a curved line. But if the body was in uniform motion, the nature of the graph would be a straight line. This straight line can be either passing from the center or a line intersecting both the axes or a line parallel to X-axis or Y-axis. Distance-time graph shows the same nature.