## Velocity - time graph and acceleration

Physics Narrative for 14-16

#### To measure velocity you need to choose a point of view

Velocity is a tricky thing to measure. One of the reasons it is tricky is that humans have no inbuilt velocity sensor. We do have inbuilt acceleration sensors, as a result of evolving and living on a mass that exerts gravitational forces on us. So one way to get a handle on velocity is to start with acceleration.

Acceleration is something we have a feel for: there are physical experiences that can be directly correlated with the physical quantity acceleration. (If you have any lingering doubts, try waggling your head, or jumping up and down, or go for a drive along a winding country lane.)

We've shown that acceleration and velocity are intimately connected. Acceleration tells velocity how to accumulate. An acceleration of 2 metre second^{-2} simply means adding 2 metre second^{-1} onto the existing velocity for each second of that acceleration. An acceleration of -2 metre second^{-2} simply means subtracting 2 metre second^{-1} from the existing velocity for each second of the acceleration. So one can see that a constant (and non-zero) acceleration will always be connected with a steady increase or decrease in velocity. If the acceleration is zero, then the velocity will neither increase or decrease: the velocity will be constant. These simple and necessary patterns allow you to make inferences in the reverse direction, from records of velocity to the acceleration connected with those velocities.

#### Graphical accumulations

One way of presenting a record of velocities is to use a scatter graph. In the case where the velocity is increasing (a positive acceleration) there will be one characteristic shape. The shape will be different if the acceleration is zero (the velocity is not changing), and different again if the velocity is decreasing (a negative acceleration).

The necessary connection between acceleration and velocity result in necessary connections between the shapes of the graphs representing a record of these values over time. So, you can move easily between the acceleration–time and velocity–time representations, so long as you remember that the connection between velocity and acceleration is that acceleration tells velocity how to change. The greater the acceleration, the greater the gradient of the velocity– time graph. If the acceleration is positive, then the gradient will also be positive (sloping upwards as time increases). If the acceleration is negative, then the gradient will also be negative (sloping downwards as time increases). If the acceleration is zero then the gradient will also be zero (no slope at all).