,
Linear Motion : Uniform Acceleration
 

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

 

 

 

 

Introduction

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 .

 

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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.

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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

rearranging

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

rearranging

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

summary:

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

 

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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

 

 

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uniform acceleration
non-uniform accln.
simple harmonic motion
projectiles
circular motion
relative motion
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elastic strings
impulse
conservation momtm
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