Question 1: Explain displacement-time graph and velocity-time graph. In each type give brief details along with appropriate diagram for illustration.

Displacement-time graph and velocity-time graph are two important tools to study the motion of an object. While drawing such graphs, the horizontal direction is specified by the independent variable and the vertical by the dependent variable. In these cases, time is the independent variable and therefore, taken along the horizontal axis.

Displacement-time graph

Graph drawn between the displacement of a body covered from a fixed point and the time taken is known as displacement-time graph. Displacement (x) is dependent variable and plotted on vertical axis, whereas, time is independent variable and plotted on horizontal axis.

Displacement-time graph gives us the important information that its slope or gradient shows the velocity of the object.

Slope of the displacement-time graph = Δy/Δx = velocity of the object.

Moreover, since displacement can be negative or positive, the graph also indicates from its slope (which can also be negative or positive) whether the body is going in the forward direction or reversed.

Analysis of the displacement-time graph

Uniform Velocity

The line of graph has throughout a constant slope. To prove this, we simply find its slope at different intervals. Let we study it for interval t1 and t2 on the horizontal axis and the corresponding values x1 and x2 on the vertical. Let the changes are Δx1 and Δt1, respectively. Then we find the changes between t3 and t4 and the corresponding values on vertical axis x3 and x4. Then we find the ratios of the changes and see that,

You can find some other results like between t2 and t3 and correspondingly on x2 and x3 and see that it also has the same constant ratio. As these ratios represent the slope of the graph (i-e, velocity), therefore, the velocity is constant throughout the motion of the body.

Uniformly increasing velocity

If the slope of the displacement-time graph is uniformly increasing, this means a uniform increase in the velocity of the body. Consider the figure below.

Determine the slope of the graph line for different intervals like, AB and CD. For interval AB, it is

For segment CD

Now, if the body is moving with increasing velocity, SlopeCD>SlopeAB. And we see it is. If the increase in velocity is uniform, the increase in the slope of the graph line will also be uniform.

Uniformly decreasing velocity

Consider the graph below. Find the slopes of the graph at different points A, B, C and D. We see that the slopes are gradually decreasing.

Or

Such a graph is the graph of decreasing velocity. If the change in slope is uniform, the velocity will also be uniformly decreasing. Moreover, the slope is negative which shows the direction of the velocity.

d-t graph for variable velocity

Consider the graph to the right.

When the motion is varying, i-e, some time the velocity is increasing and some time it is decreasing, the slope of the curve is also increasing or decreasing in the same way. AB portion of the graph shows a constantly increasing velocity. BC portion shows the slope or steepness is zero and the body is at rest. CD portion shows a negative slope slowly increasing. Hence the car has reversed its direction until it reaches to zero displacement at D.