Impulse and momentum - relation of momentum with kinetic energy, practice problems, FAQs

A truck moving on a highway is said to possess both kinetic energy (KE) and momentum due to its motion, in contrast to a bird sitting idle on a tree. Why so? Because the truck has a non-zero velocity; momentum is a measure of how fast or slow a certain amount of mass is moving times its mass; a truck having a heavier mass and moving at a speed of say , has a higher momentum compared to a car having relatively less mass and moving at a speed of . If it speeds up, then due to its increase in speed, its momentum increases, and subsequently changes the kinetic energy as well. Both the truck and the car possess KE and momentum. So what exactly are momentum and kinetic energy, and how are they related?

Table of contents

• What is Kinetic energy?
• Frame dependent nature of Kinetic energy
• What is momentum?
• Relation between KE and momentum
• What is impulse?
• Practice problems
• FAQs

What is Kinetic energy?

The energy of a body by virtue of being in motion is called kinetic energy. The KE

possessed by a truck of mass moving with velocity can be expressed as,

It is a scalar quantity. Unit of KE is Joule.

Frame dependent nature of Kinetic energy

The truck mentioned above moves in the frame of reference of ground with a velocity . Let us consider a boy riding a bicycle moving parallel to it with a velocity . A boy stands stationary and observes both of them moving.

The KE of the truck observed for different observers are as follows:

Observer	Kinetic energy
Boy riding the bicycle
Boy standing on the road

What is momentum?

Momentum( ) can be expressed as the product of mass( and velocity of the particle.

.

It is a vector quantity with unit

Relation between KE and momentum

The KE of a particle of mass and velocity can be defined as

What is impulse?

Impulse ( is used to quantify the effect of a force in a time interval which can cause momentum change in an object. If a force acting on a body of mass changes its momentum, then by Newton’s second law,

;where is the change in momentum in time interval

(final velocity,initial velocity)

∴

I.e Impulse change in momentum

Since change in momentum is a vector quantity, impulse is also a vector quantity. Its unit is or

Video explanation

From 35:02-39:36

Practice problems

Q1) If the KE of a particle is increased by , then what will be the percentage change in its momentum?

a) (b) (c) (d)

Solution (a)

Let be the new KE and be the initial KE

Then, .

Let be the initial momentum and be the final momentum.

Then,

; mass of the particle

;

∴

change in momentum .

Q2) A ball of mass and another of mass are dropped from the top of a tall building;After free falling for towards the earth, what will be the ratio of their kinetic energies?

(a) (b) (c) (d)

Solution(c)

Given and .

Let and be the respective kinetic energies of the body, and be their respective velocities.

and and ;

∴

Q3) The correct graph showing variation of and is Kinetic energy and momentum)

Solution (b)

The relation between momentum and square root of KE is .

on the axis vs on the axis is of the form

∴ It is a hyperbola.

Q4) Two masses of and are moving with equal kinetic energies. The ratio of their momenta is?

(a) : (b) (c) (d)

Solution (b)

Since , Kinetic energy, mass

FAQs

Q1.What is the difference between impulse and force?
Ans) Force is applied by an external agent to cause change in state of rest or motion of a body–it is numerically equal to the momentum change divided by time taken. However, impulse is equal to the product of force and time–a larger force acting for a larger time produces larger impulse–and possibly more damage.

Q2. Give an example of impulse and momentum?
Ans) A baseball being struck with a heavy force by a baseball bat and the contact time between them being small is an example of impulse delivered by the bat. Momentum is simply the product of the mass of baseball and its velocity and it is present in any moving body having mass.

Q3. When is the momentum of an object conserved?
Ans) As per Newton’s second law, force is equal to the change in momentum. When the net external force acting on a body is zero, then the momentum change produced is zero. For instance, when no net force is applied on the truck, it will continue to move with the same velocity–meaning there is no change in momentum of the truck.

Q4. Where is impulse encountered in everyday life?
Ans) When we strike a nail with a hammer, a huge force is applied to fit the nail in the wall. This is an example of impulse provided by hammer to nail.