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:

Concept of constant acceleration motion

The constant acceleration motion is quite common in your daily life. An object that is allowed to fall and that does not find any obstacle in its way (free fall), or a skier that descends an incline, just before arriving to the jump area, are good examples of this. The constant acceleration motion or uniformly accelerated rectilinear motion (u.a.r.m) has the following properties:

A body moves with constant acceleration motion or uniformly accelerated rectilinear motion (u.a.r.m) when its trajectory is a straight line and its acceleration is constant and different from 0. This implies that the velocity increases or decreases its magnitude uniformly.

Distance traveled in constant acceleration motion

Uniformly Accelerated Rectilinear Motion

In our example the car describes a u.a.r.m since it moves in a straight line with a constant acceleration equivalent to 2 m/s2. [Notice that, in each second, the velocity and the distance traveled by the body increase based on the value of the acceleration in the previous second.]

Notice that although colloquially, we make distinction between an accelerating and a braking body, from the point of view of physics, both are uniformly accelerated rectilinear motions. The difference is that while one has positive acceleration, the other one has negative acceleration.

Constant acceleration motion equations

The equations of the constant acceleration motion or uniformly accelerated rectilinear motion (u.a.r.m.) are:

v=v0+at

x=x0+v0t+12at2

a=cte

Where:

  • x, x0: Position of the body at a given time (x) and at the initial time (x0). Its unit in the International System (SI) is the meter (m)
  • v,v0: Velocity of the body at a given time (v) and at the initial time (v0). Its unit in the International System is meter per second (m/s)
  • a: Acceleration of the body. Remains constant with a value different from 0. Its unit in the International System is meter per second (m/s2)
  • t: Time being studied. Its unit in the International System is the second (s)

Although the former are the main equations of the u.a.r.m. and the only ones necessary to solve the exercises, it is sometimes useful know the following expression:

v2=v02+2·a·x

The above formula allows you to relate velocity and distance traveled if the acceleration is known, and can be deduced from the previous ones, as you can see below.

v=v0+a·tx=x0+v0·t+12·a·t2t=v-v0ax=v0·t+12·a·t2x=v0v-v0a+12·a·v-v0a2;

2·a·x=v2-v02

Deduction of the constant acceleration motion equations

To deduce the equations of constant acceleration motion or uniformly accelerated rectilinear movement (u.a.r.m), it must be taken into consideration that:

  • The normal or centripetal acceleration value is zero: an=0
  • The average acceleration, instant acceleration and tangential acceleration have the same value: a=aa=at=cst

With these restrictions, we get:

aa=aaa=ΔvΔt=v-v0t-t0=t0=0x-x0tv-v0=atv=v0+at

This first equation relates velocity of a body with its acceleration at any given time and represents a straight line (v) whose slope is the same as the magnitude of the acceleration and its y coordinate at the origin is the initial velocity (v0). We need to get an equation that allows us to obtain the position. There are different methods to deduce it. We will use the mean speed theorem or Merton rule of uniform acceleration:

"A body with uniformly accelerated motion travels, in any given time, the same distance that would be traveled by a body moving with a constant velocity equal to the average velocity of the first body."

This means that

x=vat

The value of the average velocity, when the acceleration is constant, can be clearly observed in the following figure:

mean speed theorem or Merton rule of uniform acceleration

va=v+v02

If we develop the equations we have seen so far, we obtain the equation of the position in the uniformly accelerated rectilinear motion (u.a.r.m.):

x=x-x0=vat=1v+v02t=2v0+at+v02t=2v0+at2t=22v0t+at22x=x0+v0t+12at2

Where have we applied:

  1. va=v+v02

  2. v=v0+at

Finally, notice that in the previous equations the motion has been considered in the x-axis. If we move in the y-axis, for example in free fall or vertical launch motions, simply substitute x with y for the position, resulting in the following equation:

y=y0+v0t+12at2

Solved exercises worksheet

Here you can test what you have learned in this section.

A bicycle with constant acceleration

difficulty

A cyclist starts his morning ride and after 10 seconds his velocity is 7.2 km/h. At that moment, he sees a dog approaching and slows down for 6 seconds until the bicycle stops. Calculate:

a) The acceleration until he begins to slow down.
b) The braking acceleration of the bike.
c) The total distance traveled.

Formulas worksheet

Here is a full list of formulas for the section Equations of Constant Acceleration Motion. 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

Position equation of uniformly accelerated rectilinear motion - y-axis

y=y0+v0t+12at2

Acceleration equation in uniformly accelerated rectilinear motion

a=cte

Velocity equation in uniformly accelerated rectilinear motion

v=v0+at

Related sections