# Introduction to mechanics

## Introduction

Mechanics is the study of the motion of objects using equations. This can be achieved by using standard formulae or mathematical techniques such as differentiation or integration.

## Definitions

A few definitions need to be discussed.

• ${\displaystyle Displacement}$  - How far an object is from its starting position. This is given the symbol ${\displaystyle s}$
• ${\displaystyle Distance}$  - How far an object has travelled. This is not usually used in calculations.
• ${\displaystyle Initial}$  ${\displaystyle Velocity}$  - The starting rate of change of displacement. This is given the symbol ${\displaystyle u}$
• ${\displaystyle Final}$  ${\displaystyle Velocity}$  - The ending rate of change of displacement. This is given the symbol ${\displaystyle v}$
• ${\displaystyle Speed}$  - The rate of change of distance. This is not usually used in calculations.*
• ${\displaystyle Acceleration}$  - The rate of change of velocity. This is given the symbol ${\displaystyle a}$
• ${\displaystyle Time}$  - How long has elapsed from the start of the event. This is given the symbol ${\displaystyle t}$

Speed can be calculated as the absolute value of velocity, ie spd = |u| or spd = |v|

## Standard Units

The SI units (standard units) of the quantities used in this section are given below:

• ${\displaystyle Displacement}$  - metres ${\displaystyle (m)}$
• ${\displaystyle Velocity}$  - metres per second ${\displaystyle (ms^{-1})}$
• ${\displaystyle Acceleration}$  - metres per second squared${\displaystyle (ms^{-2})}$
• ${\displaystyle Time}$  - seconds ${\displaystyle (s)}$

## Motion in a straight line with constant velocity

To calculate the displacement moved by an object with constant velocity we use the following formula:

${\displaystyle s=ut}$

Example: A car is travelling at 10ms${\displaystyle ^{-1}}$ . How far will it travel in 12s?
s = 10 x 12 = 120.
Thus the car will travel 120m in 12s.

Note. This formula is based on a formula shown below.

## Newton's equations of motion

In mechanics, the following formulae are VERY important. These are:

• ${\displaystyle v=u+at}$
• ${\displaystyle s=ut+0.5at^{2}}$
• ${\displaystyle s=0.5t(u+v)}$
• ${\displaystyle v^{2}=u^{2}+2as}$

These can be used to solve any motion question where there is a constant (or assumed to be constant) acceleration. Even if the acceleration is 0.
The equation from the above section ${\displaystyle (s=ut)}$  is derived from ${\displaystyle s=ut+0.5at^{2}}$  with ${\displaystyle a=0}$  and therefore ${\displaystyle s=ut+0}$  etc.

To decide which equation to use, look at the data you are given. Write down a list of the quantities given (the s,u,v,a and t values). Then cross out the quantity that is not mentioned in the question. All you have to do now is see which equation has the 4 quantities that you have left in and then solve.

Example 1: A car is travelling at 10ms${\displaystyle ^{-1}}$  when it starts accelerating at 1.5ms${\displaystyle ^{-2}}$ . How far will it travel in 15s?
s = ?, u = 10, v =, a = 1.5, t = 15
Thus use ${\displaystyle s=ut+0.5at^{2}}$ .
s = (10 x 15) + (0.5 x 1.5 x 15²)
s = 150 + 168.75 = 318.75
Thus the car will travel 318.75m

Example 2: A cyclist is initially at rest and then accelerates to 12ms${\displaystyle ^{-1}}$  in 6s. Calculate the acceleration.
s =, u = 0, v = 12, a = ?, t = 6
Thus use ${\displaystyle v=u+at}$ .
12 = 0 + (a x 6)
12 = 6a
Thus a = 2
The cyclist is accelerating at 2ms${\displaystyle ^{-2}}$

Example 3: A van accelerates at 1.5ms${\displaystyle ^{-2}}$  over a distance of 10m to 20ms${\displaystyle ^{-1}}$ . How fast was the van travelling before it started accelerating?
s = 10, u = ?, v = 20, a = 1.5, t =
Thus use ${\displaystyle v^{2}=u^{2}+2as}$ .
20² = u² + (2 x 1.5 x 10)
400 = u² + 30
u² = 370
u = ${\displaystyle {\sqrt {370}}}$  = 19.2 (3sf)
The van was travelling at 19.2ms${\displaystyle ^{-1}}$  (3sf)

Example 4: A lorry can come to a stop from 10ms${\displaystyle ^{-1}}$  in 3 seconds. How far will it travel before it stops?
s = ?, u = 10, v = 0, a =, t = 3
Thus use ${\displaystyle s=0.5t(u+v)}$ .
s = (0.5 x 3) x (10 + 0)
s = 1.5 x 10
s = 15
The lorry will cover 15m before stopping.

## Applied force

Whenever a force is applied to an object it accelerates proportionally to the size of the force applied.

ie ${\displaystyle F\alpha \ a}$ .
and therefore ${\displaystyle F=ka}$  where k is a constant.

Via calculations the constant k, is actually the mass (m) of the object being accelerated therefore:
${\displaystyle F=ma}$ .

 Force is measured in Newtons (N) and is given the symbol F
 Mass is measured in Kilograms (kg) and is given the symbol m

Example: A freighter weighing 100000kg is accelerating at 10ms${\displaystyle ^{-2}}$ . What force is required?
Use ${\displaystyle F=ma}$
F = 100000 x 10
F = 1000000N
A force of 1000kN is required. (Where 1kN = 1000N)

## Vector motion

So far we have only really looked at motion in 1 dimension. Some times you need to look at motion in 2 and maybe 3 dimensions.
To achieve this, all you have to do is convert the equations of motion into vector equations. Therefore the equations of motion become:

• ${\displaystyle {\underline {v}}={\underline {u}}+{\underline {a}}t}$
• ${\displaystyle {\underline {s}}={\underline {u}}t+0.5{\underline {a}}t^{2}}$
• ${\displaystyle {\underline {s}}=0.5t({\underline {u}}+{\underline {v}})}$

Where the underlined quantities are vector quantities none-underlined quantities are scalar.

Note: the last equation of motion is missing as this requires a more complex method of solving as it involves vector products.

Example: A ball is moving with an initial velocity of ${\displaystyle ({\underline {i}}+3{\underline {j}}-6{\underline {k}})}$ ms${\displaystyle ^{-1}}$  for 2 seconds. Assume the only acceleration is that of gravity, which can be taken to be 9.8ms${\displaystyle ^{-2}}$ . Find the position vector of the ball at time t = 2s. (i, j and k represent the x,y and z axis of the cartesean system)

${\displaystyle {\underline {s}}}$  = ?
${\displaystyle {\underline {u}}=({\underline {i}}+3{\underline {j}}-6{\underline {k}})}$
v =
${\displaystyle {\underline {a}}=-9.8{\underline {j}}}$
${\displaystyle t=2}$
Thus use ${\displaystyle {\underline {s}}={\underline {u}}t+0.5{\underline {a}}t^{2}}$
${\displaystyle {\underline {s}}=2{\underline {i}}+6{\underline {j}}-12{\underline {k}})-39.2{\underline {j}}}$
${\displaystyle {\underline {s}}=2{\underline {i}}-33.2{\underline {j}}-12{\underline {k}}}$
Thus the ball is at ${\displaystyle (2{\underline {i}}-33.2{\underline {j}}-12{\underline {k}})}$

To find the distance of the position vector, take the modulus (|a|) of the vector using Pythagoras' theorem.

## Mechanics using calculus

More often than not you will be presented with a problem that gives you the displacement/velocity/acceleration as a function rather than just values. To solve these you must use differentiation and integration techniques.

Acceleration, velocity and Displacement are related as follows:

• ${\displaystyle Displacement=\int (Velocity)dt}$
• ${\displaystyle Velocity=\int (Acceleration)dt}$

And therefore..

• ${\displaystyle Acceleration={\frac {d}{dt}}(Velocity)}$
• ${\displaystyle Velocity={\frac {d}{dt}}(Displacement)}$

Example:
A car's acceleration describes the function ${\displaystyle a={\frac {4t^{3}}{3}}-{\frac {3t^{2}}{2}}+2}$ . State the function of the velocity given that the car was at rest before accelerating.
${\displaystyle a={\frac {dv}{dt}}={\frac {4t^{3}}{3}}-{\frac {3t^{2}}{2}}+2}$
Therefore ${\displaystyle v=\int ({\frac {4t^{3}}{3}}-{\frac {3t^{2}}{2}}+2)dt}$
${\displaystyle v={\frac {t^{4}}{3}}-{\frac {t^{3}}{2}}+2t+C}$  where C is the constant of integration[/itex]
At time t = 0, v = 0. Therefore:
${\displaystyle 0=0-0+0+C\Rightarrow \ C=0}$
${\displaystyle v={\frac {t^{4}}{3}}-{\frac {t^{3}}{2}}+2t}$

## Questions on introduction to mechanics

### Questions

1. A ball is rolled 15m in 10s. How fast was it travelling?
2. A Car accelerates from rest to 25ms${\displaystyle ^{-1}}$  while travelling 15m. How long did this take?
3. An athlete accelerates from rest to 10ms${\displaystyle ^{-1}}$  in 1s. State the acceleration.
4. Calculate the acceleration created when a 10kg block has a force of 200N exerted on it.
5. Using the acceleration found above, how fast will a object travelling at 10ms${\displaystyle ^{-1}}$  be travelling after 10 seconds?
6. A rock is thrown with velocity ${\displaystyle {\underline {i}}-5{\underline {j}}-{\underline {k}}}$ . Assuming that the only acceleration acting upon the rock is gravity, (9.8ms${\displaystyle ^{-2}}$ ) state the position vector after 13s and therefore the distance after 13s.
7. An object's path is described by the function ${\displaystyle t^{2}-16t+64}$ . Find the velocity function and therefore the acceleration applied to the object.

1. 1.5ms${\displaystyle ^{-1}}$
3. 10ms${\displaystyle ^{-2}}$
4. 20ms${\displaystyle ^{-2}}$
5. 210ms${\displaystyle ^{-1}}$
6. ${\displaystyle {\underline {s}}=13{\underline {i}}-403.1{\underline {j}}-13{\underline {k}}}$ . And therefore the distance is 403.52m (2dp)
7. ${\displaystyle v=(2t-16)ms^{-1}}$  and therefore ${\displaystyle a=2ms^{-2}}$ .