Constant acceleration formulae: SUVAT
In a nutshell
The formulae derived from the velocity-time graph: v=u+at and s=(2u+v)t, can be used to create three more formulae. These formulae are essential for solving problems about objects moving in a straight line with constant acceleration. There are five formulae in total.
Equations
description | Equation |
---|
Velocity of an object with constant acceleration. | |
Displacement of an object over a certain time. | s=(2u+v)t |
Velocity of an object with constant acceleration. | v2=u2+2as |
Displacement of an object with constant acceleration using initial velocity. | s=ut+21at2 |
Displacement of an object with constant acceleration using final velocity. | s=vt−21at2 |
Variable definitions
QUANTITY NAME | SYMBOL | UNIT NAME | UNIT |
---|
Displacement | | | |
Initial velocity | | Metres per second | |
Final velocity | | Metres per second | |
Acceleration | | Metres per second squared | |
| | | |
Deriving formulae
To derive the new formulae, you can eliminate variables from the previously derived formulae.
First eliminate t.
To do this, first identify a formula to use:
v=u+at
Rearrange to make t the subject:
t=av−u
Now bring in the other derived formula:
s=(2u+v)t
Substitute t with the rearranged expression:
s=(2u+v)(av−u)=2a(u+v)(v−u)
Multiply both sides by 2a:
2as=(u+v)(v−u)
Expand the brackets:
2as=v2−u2
Rearrange to make v2 the subject:
v2=u2+2as
Next, eliminate v.
Using a previously derived formula:
v=u+at
Substitute this new expression for v into the formula for displacement and simplify:
s=(2u+v)t
s=(2u+u+at)t
s=(22u+2at)t
s=(u+21at)t
s=ut+21at2
Finally, eliminate u.
Use a previously derived formula and make u the subject:
v=u+at, therefore u=v−at
Substitute this into the new displacement equation and simplify:
s=ut+21at2
s=(v−at)t+21at2
s=vt−at2+21at2
s=vt−21at2
SUVAT
These five formulae make up the 'SUVAT' formulae. These formulae are known as constant acceleration formulae. The formulae are:
|
s=(2u+v)t |
v2=u2+2as |
s=ut+21at2 |
s=vt−21at2 |
You should be able to derive and use each of these formulae for problems about objects moving in a straight line with constant acceleration.
Example
A particle moves in a straight line from A to B with a constant acceleration of 4ms−2. The velocity of the particle at point A is 2ms−1 and the velocity at B is 20ms−1.
a: Find the distance from A to B.
b: Find the time it takes the particle to go from A to B.
First, identify the variables which are given: u=2ms−1, v=20ms−1, a=4ms−2.
a: The distance from A to B is also the displacement. The question does not give a value for t, therefore use a formula which does not include it:
v2=u2+2as
Substitute known values:
202=22+2(4)s
400=4+8s
Rearrange to make s the subject and solve:
8s=396
s=8396=49.5
The distance from A to B is 49.5 m.
b: To avoid any potential errors in the working out for part a, use the equation which does not involve s:
v=u+at
Substitute known values:
20=2+4t
Rearrange to make t the subject and solve:
4t=18
t=418=4.5
The particle takes 4.5 s to travel from A to B.