1. Constant Acceleration
Acceleration is how quickly the velocity is changing - the change in velocity per unit time. If the acceleration is constant, this can then be expressed as the following equation:
Acceleration = Change in Velocity
Time
a = (v - u) t
Change in velocity =
Final velocity - Initial velocity
Where:
a is acceleration
v is the final velocity
u is the initial velocity
t is time taken

2. Constant Acceleration
From...
a = (v - u) t
We rearrange the formula to make v the subject...
at = v - u
(multiply both sides by t)
at + u = v
(add u to both sides)
Rewriting, gives...
v = u + at

3. Calculating the Distance Travelled
Consider an object accelerating from an initial velocity u to a final velocity v in t seconds...
As before, to calculate the distance travelled, we need to find the area underneath the graph...

4. Calculating the Distance Travelled
at = v - u
a = (v - u)
From:
Area of Lilac Rectangle
= Base x Height
= t x u
= ut
Area of Blue Triangle
= ½ Base x Height
= ½ x t x (v - u)
= ½ x t x (at)
= ½ at2
s = ut + ½ at2
Total Area = ut + ½ at2 so the formula for distance travelled (s), is…

5. Average speed = Total distance travelled
We also know that the average speed can be calculated from...
Average speed = Total distance travelled
Total time taken
u + v = s t 2
If we multiply both sides of the equation by t we can find out s...
s = (u + v) x t 2
This formula allows us to calculate the distance travelled if we know the initial and final velocities and the time taken.

6. Constant Acceleration Formulas
We now have 3 constant acceleration formulas…
v = u + at
s = ut + ½ at2
s = (u + v) x t 2
We can combine formulas and by eliminating t...
The first step is to make t the subject of formula
v = u + at
v - u = at
(subtracting u from both sides)
v - u = t a
(dividing both sides by a)
t = v - u a
(rewriting)

7. Constant Acceleration Formulas
The second step is to substitute this value for t into formula
s = (u + v) x t 2
t = v - u a
s = (u + v) x 2
(v - u) a
Multiply both sides by 2a...
2as = (u + v) (v - u)
Multiply out the brackets...
So the fourth constant acceleration formula is:
v2 = u as
2as = v2 - u2
Add u2 to both sides and rewrite...
v2 = u as

8. Constant Acceleration Formulas - Summary
We now have 4 formulas. These formulas will help you calculate any motion problem in which a body undergoes zero or constant acceleration.
Symbol
Meaning
Unit v
final velocity
m/s u
initial velocity s
distance travelled m t
time taken a
acceleration
m/s2
Formulas
s = (u + v ) x t
s = ut ½at2
v = u at
v2 = u as
Whenever you have any 3 of the five ‘v-u-s-t-a’ unknowns, you can find out the remaining 2 unknown values by using one or more of the above formulas…

9. Using Constant Acceleration Formulas
s = (u + v ) x t 2
s = ut ½at2
v = u at
v2 = u as

10. Projectile Motion - Forces Acting
Ignoring air resistance, the only force acting on a projectile during the flight is gravity.
Projectiles have a downward acceleration (due to gravity) and this only affects the vertical velocity.
For a projectile there is no acceleration in the horizontal direction.
Horizontal
Vertical

11. Altering Projectile Velocity
A cannon ball is fired horizontally at a target. Can you find out (or calculate) the correct velocity to hit the target?

12. Calculating Time Taken
Example: Calculate the time taken, from firing, for the cannon ball to hit the target.
t = d/s is a formula that can be applied to solve problems, whenever velocity is constant...
Velocity is constant in the horizontal vector...
t = d/s
t = 48/24
t = 2
Time taken is 2s

13. Calculating Final Velocity
v = u + at is a formula that can be applied to solve problems, whenever acceleration is constant...
Example:
Calculate the final vertical velocity of the cannon ball as it hits the target. In this case a = g = 9.8ms-2 (9.8m/s2)
v = u + at
v = x 2
v =
v = 19.6m/s
Final vertical velocity is 19.6m/s

14. Kinetic Energy 2 starting equations Force = m x a (Newton’s 2nd law)
W = Force x distance
Together give
Work = (m x a) x distance
W = m x a x d

15. Work is MAD Av Speed (velocity) is distance time
Distance = Average velocity x time
Distance = v + u x t 2

16. Distance Distance = v + u x t 2 D = ½ x v + u x t
Starting velocity is zero
D = ½ x v x t

17. Time Final velocity = acceleration x time Time = final velocity
t = v a

18. Going back a step…..
D = ½ x v x t
D = ½ x v x v a
D = ½ x v2

19. Put into previous statements
W = m x a x d
W = m x a x ½ x v2 a
W = m x ½ x v2
W = ½ x m x v2

20. GPE work done = Force x distance Force = mass x acceleration
Gravitational constant (g) is the cause of the acceleration
Force = mass x gravitational constant
F = m x g
Work = mass x acceleration x distance
In this case height is the distance
therefore, PE=mgh

21. Work Work = force x distance Measured in Joules.
Weight is a force, mass is not

22. Power
Power = work / time (s)
Measured in Joules/second or Watts

23. Newton’s 2nd law Force = mass x acceleration Force in Newton’s
Mass in kg
Acceleration in m/s2