# Velocity-Time Graph

In previous lesson, we have learned how to analyse a displacement-time graph and finding velocity from a displacement-time graph, where the velocity is equal to the gradient of the graph.

In this lesson, we are going to proceed to velocity-time graph, where we will learn how to analyse a velocity-time graph, how to solve the calculation questions related to velocity-time graph and how to convert a velocity-time graph to other graphs.

A velocity-time graph illustrates how the velocity of a moving object varies over time. It is the most useful graph for analysing motion. In SPM, more than 90% of the questions related to graph of motion in the SPM exam are questions of velocity-time graph.

There are a few things that you need to know about a velocity-time graph. First, it tells the velocity at any instant. For example, in this graph, we can tell that the velocity of trolley B at 20s is 10 m/s whereas the velocity of trolley A at 20s is 30 m/s.

Second, the gradient of a velocity-time graph represents the acceleration of the motion. Third, the area between the graph line and the x-axis represent the displacement of the motion. Forth, the positive and negative value of the velocity tells the direction of motion. In a velocity time graph, the gradient of the graph represents the acceleration. The higher the gradient, the higher the acceleration. The lower the gradient, the lower the acceleration. For example, in this graph, the acceleration of trolley A is higher than trolley B because the gradient of the graph line of trolley A is higher than the gradient of trolley B.

If the gradient is negative, the acceleration is also negative. A negative acceleration can mean deceleration of accelerates in the negative direction.

Take notes that a velocity-time graph is different from a displacement-time graph. In displacement-time graph, the gradient is equal to velocity, whereas in velocity-time graph, the gradient is equal to acceleration.

Let’s try to find the acceleration of trolley A. The change of velocity is 30 ms-1 and the change of time is 20s. Therefore, the gradient is 30/20, equal to 1.5. The acceleration of trolley A is 1.5 ms-2.

In a velocity-time graph, the area between the graph and the x-axis represents the displacement. One thing we need to know about the area in graph is: the area above x-axis is positive whereas area below x-axis is negative. For example, in this graph, area P is 100 m2 whereas area Q is -25m2 but not +25m2. When the area is negative, the displacement must also be negative. A negative displacement means the motion is in the opposite direction.

From 0-10s, the displacement is +100m, shows that the object moved 100m in the positive direction. After 10s, the displacement become -25m, suggests that the object turned back and moved 25m in the negative direction. As a result, the total displacement is 75m.

How about the distance travelled? Distance is a scalar quantity, it has no negative value. Even though the area below the x-axis is negative, we still take the distance travelled as +25m, but not -25m. As a result, the total distance travelled is equal to 100m + 25m, equal to 125m.

This shows how we find displacement and distance from a velocity-time graph.