Linear acceleration

On this page we will consider the gradient of and area under velocity-time graphs and the equations that can be derived from these.


Key Concepts

Acceleration

Acceleration is defined as the rate of change of velocity:

\(a={v-u \over t}\)

The SI derived unit of acceleration is m s-2.

All of the equations of motion require constant acceleration.

If the acceleration is constant, average velocity equals the average of the initial and final values. This leads to a suvat equation:

\(s ={(u + v) \over 2}t\)

Speeding up or slowing down?

Acceleration is a vector quantity. In 1-dimension, its sign tells us the direction. A negative acceleration could mean:

  • slowing down in the positive direction
  • getting faster in the negative direction

Essentials

Velocity-time graphs

Gradient of a velocity-time graph

The gradient of a velocity-time graph = Δv/Δt, this is the acceleration:

\(a={v-u\over t}\)


Area of a velocity-time graph

We can see from the units m s-1 x s that, dimensionally, the area under a velocity - time graph is the displacement travelled (HINT: rotated trapezium):

\(s={(u+v)\over2} t\)

NB: We noted this equation earlier, based on taking an average of the velocities.



Displacement-time graphs

Gradient of a displacement-time graph

The displacement-time graph for motion with constant acceleration is parabolic (NB: includes a \(t^2\) term). The equation of the line is

\(s = ut + {1 \over 2} at^2\)

The gradient gives the instantaneous velocity.

Summary

Collectively these equations are referred to as suvat because of the quantities involved:

  • s = displacement /m
  • u = initial velocity /m s-1
  • v = final velocity /m s-1
  • a = acceleration /m s-2
  • t = time /s

Test Yourself

Use quizzes to practise application of theory. 


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