Linear Motion : Uniform Acceleration

[disp.-time graphs][speed-time graphs][equations][gravity]






To understand this section you must remember the letters representing the variables:

u - initial speed
v - final speed
a - acceleration(+) or deceleration(-)
t - time taken for the change
s - displacement(distance moved)

It is also important to know the S.I. unitsLe   Système International   d'Unités) for these quantities:

u - metres per second (ms-1)
v - metres per second (ms-1)
a - metres per second per second (ms-2)
t - seconds (s)
s - metres (m)

in some text books 'speed' is replaced with 'velocity'. Velocity is more appropriate when direction is important.

Displacement-time graphs

distance time graph

For a displacement-time graph, the gradient at a point is equal to the speed .


back to top


Speed-time graphs

speed time graph

For a speed-time graph, the area under the curve is the distance travelled.

The gradient at any point on the curve equals the acceleration.

acceleration as a derivative

Note, the acceleration is also the second derivative of a speed-time function.

back to top


Equations of Motion

One of the equations of motion stems from the definition of acceleration:

acceleration = the rate of change of speed

equation definition for acceleration


v equals u plus at

if we define the distance 's' as the average speed times the time(t), then:

distance equals average velocity times time


u plus v equals 2s divided by t 

rearranging (i

v minus u equals at

subtracting these two equations to eliminate v

derivation of s=ut+half at squared

it is left to the reader to show that :

v squared minus u squared equals 2as

hint: try multiplying the two equations instead of subtracting


equation summary


Example #1

A car starts from rest and accelerates at 10 ms-1 for 3 secs.
What is the maximum speed it attains?

linear horizontal motion problem #1

Example #2

A car travelling at 25 ms-1 starts to decelerate at 5 ms-2.
How long will it take for the car to come to rest?

linear horizontal  motion problem#2


Example #3

A car travelling at 20 ms-1 decelerates at 5 ms-2.
How far will the car travel before stopping?

linear horizontal motion problem #3

Example #4

A car travelling at 30 ms-1 accelerates at 5 ms-2 for 8 secs.
How far did the car travel during the period of acceleration?

linear horizontal motion problem #4


back to top


Vertical motion under gravity

These problems concern a particle projected vertically upwards and falling 'under gravity'.

In these types of problem it is assumed that:

air resistance is minimal

displacement & velocity are positive(+) upwards & negative(-)downwards

acceleration(g) always acts downwards and is therefore negative(-)

acceleration due to gravity(g) is a constant


Example #1

A stone is thrown vertically upwards at 15 ms-1.

(i) what is the maximum height attained?
(ii) how long is the stone in the air before hitting the ground?

(Assume g = 9.8 ms-2. Both answers to 2 d.p.)

gravity problem #1a

gravity problem #1b


Example #2

A boy throws a stone vertically down a well at 12 ms-1.
If he hears the stone hit the water 3 secs. later,

(i) how deep is the well?
(ii)what is the speed of the stone when it hits the water?

(Assume g = 9.8 ms-2. Both answers to 1 d.p.)

gravity proble #2a

gravity problem #2b



back to top


your stop for the best in math, science & programming tutorials on the Net revision help to get a better result incremental success advanced physics for secondary/high school, including much in-depth content common to first year university courses your one stop for the best in math, science and programming tuition revision help for a better result incremental success advanced physics for high school/secondary and 1st year university fast-track learning for everyone




linear motion
average velocity
instantaneous velocity
circular motion 1
projectile motion
parabolic motion
relative motion
Newton's 2nd Law
connected particles
KE & PE changes
elastic strings
conservation momtm.
coefft. restitution




EdxlM1 Mechanics spec.
EdxlM1 Mechanics ans.
EdxlM2 Mechanics spec.
EdxlM2 Mechanics ans.
EdxlM3 Mechanics spec.
EdxlM3 Mechanics ans.
EdxlM4 Mechanics spec.
EdxlM4 Mechanics ans.
EdxlM5 Mechanics spec.
EdxlM5 Mechanics ans.


uniform acceleration
non-uniform accln.
simple harmonic motion
circular motion
relative motion
Newton's Laws
connected particles
work & energy
power & efficiency
more circular motion
elastic strings
conservation momtm
coeff. restitution
part. forces in equilib.
rigid bodies