Impulse & Momentum

Physics 11

Problem

A baseball of mass 0.145kg is pitched toward a batter with an initial velocity of 35 m/s. If the batter hits the ball in the opposite direction at 45 m/s, determine the force that is applied by the bat on the ball if the contact time was 0.013s.

Impulse

When an object is accelerated, typically the force will only be applied for a given time

So instead of considering Newton’s Second Law as we have previously discussed it, we will rearrange the equation

Impulse

Use Newton’s Second Law and substitute the definition for acceleration

Rearrange the equation so there are no terms in the denominator

This expression is known as impulse (J)

vmtFJtvmF

amF

Example

A tennis ball is struck by a racquet with a force of 750N; if the time of contact was 0.023s, what impulse was delivered to the ball?

NsJ

sNJ

vmtFJ

17

)023.0(750

Momentum

Momentum can also be defined starting from Newton’s Second Law

The rate of change of momentum can also be used to determine the force

Momentum is defined as the product of an object’s mass and velocity 1st Law of Motion

vmptvmvm

F

tvmF

amF

if

Example

A cyclist is travelling at 32km/h and the bike and rider have a mass of 85kg. What is their momentum?

skgmpsmkgp

vmp

/760)/89.8(85

Impulse-Momentum

Impulse and momentum can be related in order to solve dynamics problems

Substitute the definition for momentum into the impulse equation

ptF

pptF

vmvmtF

vmtF

if

if

Example – again…

A baseball of mass 0.145kg is pitched toward a batter with an initial velocity of 35 m/s. If the batter hits the ball in the opposite direction at 45 m/s, determine the force that is applied by the bat on the ball if the contact time was 0.013 s.

Example

NFs

smsmkgF

tvmF

ptF

890013.

)/35(/45(145.

Collisions

Physics 11

Conservation of Momentum

The vector quantity momentum will be conserved in any collision That is, the sum of all momenta prior to the

collision will be equal to the sum of all momenta following a collision

Every object that has mass and velocity will have momentum and must be included in the total momentum of the system

Collisions

With any collision, it is imperative that you diagram the system prior to and following the collision and identify all objects involved in the collision

This allows you to ensure that you calculate the total momentum for the system to properly analyze the situation

While this may seem onerous, generally we will be looking at a maximum of two particles

Recall Momentum

Momentum

Impulse

vmp

pvmtFJ

Momentum Conservation Momentum is conserved

This is an expression of Newton’s first law: “An object at rest or in uniform motion will remain at

rest or in uniform motion unless acted on by an external force.”

External forces can change the momentum of a system (Impulse)

FI pp

pvmtFJ

'pp

Momentum Conservation

In interactions between two bodies, momentum of one object can change, but the total momentum of the system remains constant.

'' 2121 pppp

'' 22112211 vmvmvmvm

Types of Momentum Problems Elastic collisions

Inelastic collisions

Explosions

Initial

Initial

Initial Final

Final

Final

Explosions: Recoil

A Barrett M82 is a high calibre sniper rifle. Below are it’s specifications: Barrel length: 73.7 cm Weight: 14.0 kg Muzzle Velocity: 853 m/s Typical ammunition weight: 50.0g

Calculate the magnitude force exerted on the riflemen.

Explosion: Recoil

The plan is this Calculate the momentum of the rifle

knowing the momentum of the bullet Calculate the impulse imparted to the

riflemen to stop the gun. Impulse is change in momentum Impulse is force multiplied by time

Need to know the time (how long) the explosion takes.

The Momentum Problem

sm

sm

vkgvkg

vmvmvmvm

vmvmvmvm

05.3')853)(0500.0()')(0.14(

''''0

''

1

1

2211

2211

22112211

Barrel length: 73.7 cm Weapon mass: 14.0 kg Muzzle Velocity: 853 m/s Typical ammunition mass: 50.0g

Impulse

tFp

pvmtFJ

We know the person stops the gun, so to find the force, we need to know the interaction time.

Explosions: Recoil Barrel length: 73.7 cm Weapon mass: 14.0 kg Muzzle Velocity: 853 m/s Typical ammunition mass: 50.0g

Mm

L

FvIv

smvvdt

tvvd

sm

sm

IF

IF

00173.000.0853)737.0(22

21

Bring it all together

tFp

smkg

sm

IfRifle

p

kgpvmp

ppp

65.42

)'05.3)(0.14(0

1

11 st 00173.0

NxtpF 41046.2

00173.065.42

For Comparison

How many people would have to sit on your shoulder to get the same force?

NxFGun41046.2

NxkgmgFsm

person2109.6)81.9)(70( 2

PersonGun FF

36

Collision

A billiard ball, mass 155g, is travelling at 3.5m/s across the table. It strikes another ball at rest, mass 150g and comes to rest. What is the velocity of the second ball after the collision?