Mechanics / Physics

Kinematics - Equations of Motion


kinematics Suppose an object starts moving at u and gains an acceleration of a. After time t, it gets a speed v after travelling a distance of s. The graph shows this data on a velocity-time grid.

a = (v - u) / t => v - u = at

v = u + at

s = ut + 1/2(v - u) t
s = ut + 1/2 (at) t = ut + 1/2 at2

s = ut + 1/2 at2

v2 = (u + at)2
v2 =u2 + 2uat + u2t2
v2 = u2 + 2a (s = ut + 1/2 at2)

v2 = u2 + 2as


E.g.1

An object starts from rest and moves with an acceleration 2 ms-2. Find its speed after 5 seconds and distance travelled.
u = 0, t = 5s, a = 2
v = 0 + 2 x 5
v = 10 ms-1

E.g.2

An object starts moving at 10 ms-1 and gains an acceleration of 2 ms-2. Find its speed after 5 seconds and the distance travelled.
u = 10, a = 2, t = 5
v = 10 + 2 x 5
v = 20 ms-1

E.g.3

An object start moving at 20 ms-1 and increases its speed to 40ms-1 in 5 seconds. Find its acceleration and the distance travelled during this time.
v = 40, u = 20, t = 5
40 = 20 + 5a
5a = 20
a = 4 ms-2
s = 20 x 5 + 1/2 x 4 x 25
s = 100 + 50
s = 150 m

E.g.4

An object starts moving at 20ms-1 and reduces speed to 10 ms-1 in 2 seconds. Find its deceleration and the distance travelled. How far further will it move before coming to a halt?
u = 20, v = 10, t = 2,
10 = 20 + 2a
2a = -10
a = -5 ms-2
s = 20 x 2 - 1/2 x 5 x 4
s = 40 - 10
s = 30 m
When it stops,
v = 0, u = 10, a = -5
0 = 100 + 2 x -5 x s
10s = 100
s = 10 m

E.g.5

An object start moving at 10 ms-1 with an acceleration 2ms-2. Calculated the distance travelled by it during the third second.
u = 10, a = 2, t = 2
s = 10 x 2 + 1/2 x 2 x 4
s = 20 + 4
s = 24 m
u = 10, a = 2, t = 3
s = 10 x 2 + 1/2 x 2 x 9
s = 20 + 9
s = 29 m
Distance travelled during the third second = 29 - 20 = 9m.

E.g.6

A ball is thrown upwards at 20 ms-1. Find the time taken by it before reaching a height of 15 m. Assume g = 10 ms-2
u = 20, a = -10 s = 15
15 = 20t - 1/2 x 10 x t2
5t2 - 20t + 15 = 0
t2 - 4t + 3 = 0
(t - 3)(t-1) = 0
t = 3 or t = 1
The object can be at the height of 15m on two occasions - while going up and coming down.

Practice is the key to mastering maths; please visit this page, for more worksheets.

Please work out the following questions to complement what you have just learnt.


    1) The speed of a car goes down from 40 ms-1 to 35 ms-1 in 1/2 seconds. Find its deceleration. How far will it further travel before stops?
    2) A car starts from rest and moves with an acceleration 4ms-2. Find its speed after 6 seconds and the distance travelled during that time.
    3) A stone is dropped into a well. Its splash could be heard after 4.25 seconds since its drop. Calculate the depth of the well, if the speed of sound is 320 ms-1 and g = 10 ms-2.
    4) A stone is dropped from the top of a tower. It goes through 16/25th of the total height during the last second. Find the height of the tower and the time taken for the fall. g = 10ms-2
    5) A balloon is ascending at a constant speed of 20 ms-1. An object falls from it after 4 s in motion. Find the distance travelled by the object before it hits the ground. g = 10 ms-2.
    6) A vehicle is moving at a constant acceleration. It passes the town P at 66 kmh-1 and town Q at 74kmh-1, a distance of 40 km between the two. Find the acceleration and the initial velocity of the vehicle. Hence fint the time taken by the vehicle to pass a town that is 10 km from P.
    7) A car accelerates at 4ms-2, starting from rest, for 5 seconds, then maintains the speed for 10 seconds, and eventually slows down to a halt in 4 seconds. Find the deceleration during the final stage and the total distance covered. What is the average speed of the car?
    8) A car starts from rest and moves with an acceleration of 4ms-2 for 20 seconds. It then comes to a halt in the next 5 seconds. Find the deceleration of the car and the distance covered during the first 22 seconds.
    9) A car has been moving at a constant speed of 20ms-1. A lorry starting from rest moves with an acceleration of 4ms-2. How long will the lorry take before it gets the same speed of the car? How far have both gone by then? Find the time taken by the lorry to catch with the car and the distance travelled by then as well.
    10) An object slows down its speed from 20ms-1 to 10ms-1 in 4 seconds. Find its deceleration and the distanced travelled. It then comes to a halt in 5 seconds. Find the total distance travelled by the object.