The uniformly accelerated rectilinear motion (u.a.r.m.), also known as constant acceleration motion, is a rectilinear motion that has a constant acceleration, which is different from zero. In this section we are going to study:

Constant acceleration motion graphs

Position-time (x-t) graph

x=x0+v0t+12at2

The graph position-time (x-t) of a constant acceleration motion, or uniformly accelerated rectilinear motion (u.a.r.m.), represents time on the horizontal axis (t-axis) and position on the vertical axis (x-axis). Observe as the position (normally the x-coordinate) increases (or decreases) uniformly with time. This happens because as time passes, the magnitude of the velocity varies. We can distinguish two cases, when the velocity is positive or when it is negative:

Position - time (x-t) graph in constant acceleration motion

Velocity-time (v-t) graph

v=v0+at

The graph velocity-time (v-t) of a constant acceleration motion, or uniformly accelerated rectilinear motion (u.a.r.m.), represents time on the horizontal axis (t-axis) and velocity on the vertical axis (x-axis). Observe as the velocity increases (or decreases) uniformly with the passage of time. This happens as the result of the acceleration. Again, we can distinguish two cases:

Velocity - time (v-t) graph in constant acceleration motion

We can get the velocity from the angle α. To do it just remember that in a right triangle the tangent of each of its angles is defined as the opposite side (cathetus) divided by the adjacent one:

tanα=opposite sideadjacent side=vt=v-v0t=a

The value of the slope is the magnitude of the acceleration. Therefore, the greater the slope of the straight line, the higher the acceleration of the body.

Notice that the area under the curve v between two instants of time, numerically matches the distance traveled. Could you tell why?

Distance traveled and area under the velocity graph in constant acceleration motion

The area under the curve can be calculated as the area of the rectangleo S1 that would correspond to a uniform rectilinear motion (u.r.m.) to which we will add the area of the triangle S2:

x=x-x0=S1+S2=1v0t+v-v0t2=2v0t+12at2

Where have we applied:

  1. S1=v0tS2=Srectangle2=v-v0t2
  2. v-v0=at

Acceleration time (a-t) graph

a=cst

The graph acceleration-time (a-t) of a constant acceleration motion, or uniformly accelerated rectilinear motion (u.a.r.m.), shows that the acceleration remains constant over time. It is the average acceleration, which in the case of u.a.r.m. is the same as the instantaneous acceleration. Again, we can distinguish two cases:

Acceleration - time (a-t) graph in constant acceleration motion

Notice that the area under the curve, enclosed between two instants of time, numerically matches the experienced increase of velocity. Do you know why?

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Formulas worksheet

Here is a full list of formulas for the section Constant Acceleration Motion Graphs. By understanding each equation, you will be able to solve any problem that you may encounter at this level.

Click on the icon   to export them to any compatible external program.

Position equation of uniformly accelerated rectilinear motion - x-axis

x=x0+v0t+12at2

Velocity equation in uniformly accelerated rectilinear motion

v=v0+at

Acceleration equation in uniformly accelerated rectilinear motion

a=cte

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