Vectors & Scalars

What is a scalar?

Scalar quantities are measured with numbers and units.

length

(e.g. 102°C)

time (e.g. 16cm)

temperature

What is a vector?

Vector quantities are measured with numbers and units, but also have a specific direction.

acceleration (e.g. 30m/s2

displacement (e.g. 200miles

Speed or velocity?

Distance is a scalar and displacement is a vector. Similarly, speed is a scalar and velocity is a vector.

Speed is the rate of change of distance in the direction of travel. Speedometers in cars measure speed.

Velocity is a rate of change of displacement and has both magnitude and direction.

average speed

average velocity

Averages of both can be useful:

distance time

displacement time

Adding vectors

Displacement is a quantity that is independent of the route taken between start and end points.

Any two vectors of the same type can be added in this way to find a resultant.

Two or more displacement vectors can be added

‘nose to tail’ to calculate a

resultant vector. A

B C

resultant vector

If a car moves from A to B and then to C, its total

Simplifying vectors

Because of the way vectors are added, it is always possible to simplify a vector by splitting it into components.

A

B C

Imagine that instead of travelling via B, the car travels via D:

D

The car has a final displacement of x miles east and y miles north. This can be represented by (x,y).

x

y

Its displacement is the same, but it is now much easier to describe.

x component y c o m p o n en t

Motion under constant acceleration

Calculating vector quantities such as velocity or displacement can be complicated, but when acceleration is constant, four equations always apply.

v

=

u + at

v

=

u

+ 2as

s

=

½(u + v)t

s

=

ut + ½at

These are sometimes known as the constant acceleration equations, or the ‘uvast’ or ‘suvat’ equations.

Representing motion

The symbol a represents acceleration. The symbol t represents time.

Velocity at time t is represented by v, and u represents the value of v when t = 0. This is the initial velocity.

 a = acceleration  t = time

 s = displacement  v = velocity

 u = initial velocity

Two velocity equations

Two of the constant acceleration equations are velocity equations.

The first of these can be used to find the velocity at a

particular time t. The second can be used to find the velocity at a particular displacement s.

When deciding which equation to use, it is good practice to write down what you know about the values of u, v, a, s and t before you start any calculations.

v

=

u

+ 2as

v

=

u + at

Calculating final velocity

A cyclist accelerates towards the end of the race in order to win. If he is moving at 6m/s then accelerates by 1.5m/s2 for the final five seconds of the race, calculate his speed as he crosses the line.

u = 6 v = ?

a = 1.5m/s2

s = t = 5s

v = 6 + (1.5 × 5)

v = u + at

v = 13.5m/s

First write down what you know about u, v, a, s and t:

Rearranging an equation – calculations

A coin is dropped from a window. If it hits the

ground at 10m/s, work out the height of the window.

u = 0m/s v = 10m/s a = 10m/s2

s = ? t =

s = 5m

The question involves the

variables u, v, a and s, so the

relevant equation is v2 = u2 + 2as.

s = s = s = 2a

v2 – u2

2 x 10 102 – 02

20 100

First write down what you know about u, v, a, s and t:

Start by rearranging the

Two displacement equations

Two of the constant acceleration equations are displacement equations.

The first of these gives the displacement as a function of the initial velocity and velocity at time t. The second gives it as a function of initial velocity, acceleration and time.

As with the velocity equations, it is good practice to write down what you know about the values of u, v,

a, s and t before you attempt any calculations.

u = ? v = ? a = ? s = ? t = ?

Calculating displacement

A car travelling at 20m/s takes five seconds to stop. What is the stopping distance of the car?

u = 20m/s v = 0m/s a =

s = ? t = 5s

s = 50m

s = ½ × 20 × 5 s = ½ × (20 + 0) × 5

First write down what you know about u, v, a, s and t:

Rearranging an equation – calculations

Calculate the acceleration of an ambulance if it starts at rest and takes six seconds to travel 50m.

u = 0m/s v =

a = ?

s = 50m t = 6s

a =

a = 2.8m/s2

50

a =

18 a =

50 – (0 × 6) ½t2

s – ut

First write down what you know about u, v, a, s and t:

The question involves the

variables u, a, s and t, so the

relevant equation is s = ut + ½at2.

Start by rearranging the

Describing parabolic motion

 A trajectory is the path of any moving object.

 A projectile is an object that is given an initial force, then allowed to move freely through space.

 A parabola is the name given to the shape of the curve a projectile follows when gravity is the only force acting on it.

Forces on a projectile

What are the two forces that act on a freely moving projectile?

Air resistance and gravity.

Depending on the shape and density of an object, it is often possible to ignore the effects of air resistance.

This reduces the forces on the object to one constant force in a constant direction, giving:

Horizontal motion is therefore very simple, and vertical motion can be solved with the constant acceleration equations.

Solving projectile motion

How should you go about solving a problem involving projectile motion? Split the problem into two parts:

Horizontal motion at constant velocity:

 Use the equation, speed = distance / time (equation 1).

Vertical motion under constant acceleration:

 As always, start by writing down what you already know about u, v, a, s and t.

 Choose a constant acceleration equation (equation 2).

Momentum

Momentum is a vector quantity given by the following formula:

To have momentum, an object needs to have mass and to be in motion.

momentum = mass × velocity

Which of these two vehicles do you

Conservation of momentum

initial

momentum

final

momentum

=

Momentum is always conserved in any event or interaction:

When a snooker ball collides with another, what happens to the momentum of each ball?

What happens when a car brakes and comes to a halt, or when a rock falls to the ground and stops?

Momentum is transferred to the Earth.

Explosions and recoil

Remember: momentum is always conserved.

Calculating momentum

Momentum is a vector quantity. The van has a momentum of 1500 × 10 = 15000 kgm/s to the right. The car’s momentum is 15000kgm/s to the left, or –15000kgm/s to the right:

10m/s 10m/s

Two vehicles, each weighing 1.5tonnes, are driving towards each other at 30mph. If they collide, what will happen?

Both vehicles come to a halt. What has happened to their momentum?

initial momentum to the right = 15000 + –15000 = 0