How to Understand Kinetic Energy, Momentum and Work Done

REVIEWED BY
Leonard Kelley, Bachelors in Physics and Mathematics

What Is Momentum?

Momentum is the product of the mass and velocity of a body.

If m is the mass of a body and v is its velocity, then:

momentum = mv

So, the heavier an object is, the greater its momentum when it's moving if its velocity doesn't change. If it moves faster, its momentum also increases, even if its mass doesn't change.

In a collision between two or more bodies, momentum stays the same. This is known as the conservation of momentum. This means that the total momentum of the bodies before the collision equals the total momentum of the bodies after the collision.

So if m₁ and m₂ are two bodies with velocities of u₁ and u₂ respectively before the collision and velocities of v₁ and v₂ after the collision, then the sum of momentum of the two bodies before the collision equals the sum of momentum of the bodies after the collision:

m₁u₁ + m₂u₂ = m₁v₁ + m₂v₂

Elastic and inelastic collisions

Elastic collisions

In the case of a fully elastic collision (or almost totally elastic such as when two rigid snooker balls collide) kinetic energy is also conserved and not transformed into another form of energy such as heat. If an elastic body collides with a second stationary body, the momentum of the moving body is transferred to the stationary body and the first body becomes stationary.

Inelastic collisions

In this case, kinetic energy is not conserved and some of it is converted into heat, light or sound and deformation of the body. An example of a partially inelastic collision is when two pieces of modeling clay collide.

Example

Two bodies with mass 5 kg and 2 kg and velocities 6 m/s and 3 m/s respectively collide. After the collision the bodies remain joined. Find the velocity of the combined mass.

This is an example of a perfectly inelastic collision.

Recommended

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Let m₁ = 5 kg

Let m₂ = 2 kg

Let u₁ = 6 m/s

Let u₂ = 3 m/s

m₁u₁ + m₂u₂ = m₁v₁ + m₂v₂

Since the bodies are combined after the collision, v1 = v2. Let's call this velocity v.

So:

m₁u₁ + m₂u₂ = m₁v₁ + m₂v_{2 =} m₁v+ m₂v = (m₁ + m₂)v

Substituting:

(5)(6) + (2)(3) = (5 + 2)v

30 + 6 = 7v

So v = 36/7

What Is Kinetic Energy?

Kinetic energy is a related concept to that of momentum. A body has kinetic energy when it is in motion.

If a body of mass m is moving at a velocity v, then:

Kinetic energy = (1/2)mv²

The kinetic energy of a body at velocity v is the work that must be done on the body to accelerate it to that velocity.

Example

A rifle bullet of mass 4 grams is moving at a velocity of 1200 m/s.
Calculate its energy.

First convert to SI units, so 4 g = 0.004 kg

Kinetic energy = (1/2)mv² = (1/2)0.004(1200)² = 2880 joules

To put this into perspective, this is approximately the same impact energy that a 34kg (75 pound) weight would produce if dropped from chimney height off a two storey house. The bullet has a small mass, but the squared term in the equation increases the energy massively. Every time velocity is doubled, energy increases fourfold. If velocity is tripled, energy increases ninefold. An asteroid impacting on Earth can create so much destruction, not necessarily because of its mass, but because of its extremely high velocity that can be tens of thousands of kilometres per hour.

What Is Work?

We learned about forces in two previous tutorials "What Is a Force? Mass, Velocity, Acceleration and Adding Vectors" and "Newton's 3 Laws of Motion: Force, Mass and Acceleration". The definition of work in physics is that "work is done when a force moves a body through a distance". If there is no movement of the point of application of a force, no work is done. So for instance, a crane that is simply holding a load at the end of its steel rope is not doing work. Once it starts hoisting the load, it is then doing work. When work is done there is energy transfer. In the crane example, mechanical energy is transferred from the crane to the load, which gains potential energy because of its height above the ground.

The unit of work is the joule.

If work done is W

distance is s

and the force applied is F

then

W = FsCos Θ where Θ is the angle between the force and displacement

Example

A force F = 50 N is applied to a box of mass 4 kg resting on the ground. Friction between ground and box results in a force opposing motion which is Fₛ = 2 N. Calculate the acceleration and work done sliding the box 3 m.

Start by writing the force equation. Use Newton’s Second Law. The sum of forces produces a net acceleration.

We can use a convention for writing the force equation. So consider the forces acting “forwards” and accelerating the box as positive and forces acting to decelerate the box as negative, Since the friction force Fₛ acts opposite to the direction of motion, its sign is negative, so Newton's second law says that the sum of forces equals the mass multiplied by the acceleration of a body and:

F + (- Fₛ ) = ma

Substituting for F and Fₛ in the first equation gives:

50 + (- 2) = 50 - 2 = 4 x a

Rearranging:

a = (50–2) / 4 = 12 m/s²

Work done on the box is the applied force multiplied by the distance. Work is done opposing friction plus accelerating the box. The net force accelerating force is 50 - 2 = 48 N. The displacement is in the direction of the force, so Θ = 0 and Cos Θ = 1, so:

Work done = FCos Θs = 50 x 1 x 3 = 150 joules

Note however that although the work done is due to the 50 N force, the net force that accelerates the box is 50 - 2 = 48 N.

References

Hannah, J. and Hillerr, M. J., (1971) Applied Mechanics (First metric ed. 1971) Pitman Books Ltd., London, England.

This content is accurate and true to the best of the author’s knowledge and is not meant to substitute for formal and individualized advice from a qualified professional.

© 2022 Eugene Brennan