Lesson 9, Topic 1
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# Linear Momentum

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### Momentum:

Momentum of a body is the product of the mass of a body and its linear velocity. The symbol that is used to represent momentum is p.

Momentum, p = mass (m) x velocity (v)

p = kg x ms-1.

Its unit is kgms-1. It is a vector quantity.

The momentum of a body depends on the mass and the velocity with which the body is moving. Its direction is the same as the direction of the object’s velocity.

A bus can have a large momentum even if it’s moving slowly because of its large mass.

A bullet can have a large momentum even though it has a small mass because it’s moving with a high velocity.

If an object is at rest it has no momentum no matter how large its mass is.

In this case mass x 0 velocity = 0 Momentum

Note: If you want it in terms of Newton it becomes ‘newton second’.

Here is why;
Force = mass × acceleration (a)
Unit of acceleration = m/s2
We know that unit of force is Newton (N) and
1 N = $$\frac{kgm}{s^2}$$ ….(1)
Momentum (p) = mass (m) × velocity (v)
Taking the SI units
Unit of mass = kg
Unit of velocity = m/s
So
P = $$\frac{kgm}{s}$$
Suibstitute in equation (1)
F = N = $$\frac{kgm}{s^2}\\ = \frac{kgm}{s \: \times \: s}\\ = \frac{p}{s}$$
if F = $$\frac{p}{s}$$
Then, Momentum, p = Ns
p = Newton × second

### Example 1:

A car of mass 145 g moves with a velocity of 30m/s. Calculate the momentum of the car.

Solution:

Values given:

Mass = 145 g =  $$\frac{145}{1000}$$ = 0.145 kg
velocity = 30 m/s
momentum = ?

Momentum, p = m x v

p = 0.145 x 30 = 4.35 Nm

Momentum = 4.35 kgm/s

### Example 2:

Given that the momentum of a train is 354 kgm/s, calculate the mass if its velocity is 17.6 m/s.

Soultion:

Values given:

Momentum = 354 kgm/s
velocity = 17.6 m/s
mass = ?

Momentum, p = mass x velocity

make mass the subject of the formula

mass = $$\frac{momentum}{velocity}$$

mass = $$\frac{354}{17.6}$$

mass = 20.11kg

### Example 3:

A train of mass 6000kg moves with a velocity V. If the momentum of the train is 1000Nm, calculate the velocity of the train.

Solution:

Values given:

Momentum = 1000 Nm
velocity = ?
mass = 6000kg

Momentum, p = mass x velocity

make velocity the subject of the formula

velocity = $$\frac{momentum}{mass}$$

Substitute in the values

velocity = $$\frac{1000}{6000}$$

velocity = 0.167 m/s

### Example 4:

A body of mass 20kg moves with a velocity of 10ms-1, calculate its linear momentum

Solution:

Values given:

mass = 20kg
velocity = 10ms-1

Momentum = mass x velocity

= 20kg x 10ms-1

Momentum = 200kgms-1.

### Impulse:

The impulse of a force is the product of average force and the time during which it acts on the body. A force which acts for a very short time is called impulsive force or simply impulse. Impulse is usually associated with collision and is also called a blow

Impulse = Force x time

I = F x t

Impulse = Newton x seconds

I = Ns

The unit of impulse is Ns. It is a vector quantity.

### Example 5:

A force of 40N acts on a body for 5 seconds. What is the impulse experienced?

Solution:

Values given:

F = 40N, t = 5 secs.

Impulse = Force x time

Impulse = 40 x 5

Impulse = 200Ns.

### Example 6:

If a force of 45 N acts on 10.75 kg mass and the impulse is 44.94Ns, calculate the time for which the force acts on the object.

Solution:

Values given:

Force = 45N
mass = 10.75kg
Impulse = 44.94Ns

Impulse = Force x time

make time the subject of the formula

time = $$\frac{Impulse}{Force}$$

time = $$\frac{44.94}{45}$$

time = $$\frac{44.94}{45}$$

time = 0.998 = 1s

### Example 7:

The impulse of a force is 458 Ns. If the force acts for 1465 seconds, calculate the magnitude of the force.

Solution:

Values given:

Force = ?, t = 1465 seconds, Impulse = 458 Ns

Impulse  = Force x time

make Force the subject of the formula

Force = $$\frac{Impulse}{time}$$

Force = $$\frac{458}{1465}$$

Force = 0.312N

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