PHYSICS 231

INTRODUCTORY PHYSICS I

PHYSICS 231

INTRODUCTORY PHYSICS I

Lecture 8

• Work for nonconstant force

• Spring force

• Potential Energy of Spring

• Power

Last Lecture

F =−kx

PE =12

kx2

€

P =ΔW

Δt=

ΔKE

Δt

€

P = Fv

Fx

x

Chapter 6

Momentum and Collisions

Momentum

Definition:

Newton’s 2nd Law:

rp =m

rv

€

rF = m

Δr v

Δt

€

rF =

Δr p

Δt

€

⇒

Conservation of Momentum

True for isolated particles (no external forces)

Proof:

Recall F12=-F21, (Newton’s 3rd Law)

for isolated particles never changes!

€

rF 12 +

r F 21 = 0

⇒Δ

r p 1

Δt+

Δr p 2

Δt= 0

€

⇒ Δ r

p 1 + Δr p 2 = 0

r p 1 f +

r p 2 f =

r p 1i +

r p 2i

€

rp i∑

Momentum is a Vector quantity

• Both px and py are conserved

px =mvx

py =mvy

Example 6.1

An astronaut of mass 80 kg pushes away from a space station by throwing a 0.75-kg wrench which moves with a velocity of 24 m/s relative to the original frame of the astronaut. What is the astronaut’s recoil speed?

0.225 m/s

Center of mass does not accelerate

Xcm ≡m1x1 +m2x2 +m3x3 + ...(m1 +m2 +m3 + ...)

ΔXcm =m1Δx1 + m2Δx2 + m3Δx3 + ...

(m1 + m2 + m3 + ...)

= Δt ⋅m1(Δx1 / Δt) + m2 (Δx2 / Δt) + m3(Δx3 / Δt) + ...

(m1 + m2 + m3 + ...)

= Δt ⋅p1 + p2 + p3 + ...

(m1 + m2 + m3 + ...)

= 0 if totalP iszero

Example 6.2Ted and his ice-boat (combined mass = 240 kg) rest on the frictionless surface of a frozen lake. A heavy rope (mass of 80 kg and length of 100 m) is laid out in a line along the top of the lake. Initially, Ted and the rope are at rest. At time t=0, Ted turns on a wench which winds 0.5 m of rope onto the boat every second.

a) What is Ted’s velocity just after the wench turns on?

b) What is the velocity of the rope at the same time?

c) What is the Ted’s speed just as the rope finishes?

d) How far did the center-of-mass of Ted+boat+rope move

e) How far did Ted move?

f) How far did the center-of-mass of the rope move?

0.125 m/s

-0.375 m/s

0

0

12.5 m

-37.5 m

Example 6.3

A 1967 Corvette of mass 1450 kg moving with a velocity of 100 mph (= 44.7 m/s) slides on a slick street and collides with a Hummer of mass 3250 kg which is parked on the side of the street. The two vehicles interlock and slide off together. What is the speed of the two vehicles immediately after they join?

13.8 m/s =30.9 mph

Impulse

Useful for sudden changes where the exact details of the force are difficult to determine

Impulse =FΔt=Δp

For nonconstant F,Impulse = Area under F vs. t curve

Bungee Jumper Demo

Example 6.4 A pitcher throws a 0.145-kg baseball so that it crosses home plate horizontally with a speed of 40 m/s. It is hit straight back at the pitcher with a final speed of 50 m/s.

a) What is the impulse delivered to the ball?

b) Find the average force exerted by the bat on the ball if the two are in contact for 2.0 x 10–3 s.

c) What is the acceleration experienced by the ball?

a) 13.05 kgm/s b) 6,525 N c) 45,000 m/s2

Collisions

• Momentum is always conserved in a collision

• Classification of collisions:

• ELASTIC• Both energy & momentum are conserved

• INELASTIC• Momentum conserved, not energy• Perfectly inelastic -> objects stick• Lost energy goes to heat

• Catching a baseball• Football tackle• Cars colliding and sticking• Bat eating an insect

Examples of Examples of Perfectly Perfectly

Elastic CollisionsElastic Collisions• Superball bouncing• Electron scattering

Examples of Examples of Perfectly Inelastic Perfectly Inelastic

CollisionsCollisions

Ball Bounce Demo

Example 6.5a

A superball bounces off the floor,

A) The net momentum of the earth+superball is conservedB) The net energy of the earth+superball is conservedC) Both the net energy and the net momentum are conservedD) Neither are conserved

Example 6.5b

A astronaut floating in space catches a baseball

A) Momentum of the astronaut+baseball is conservedB) Mechanical energy of the astronaut+baseball is conservedC) Both mechanical energy and momentum are conservedD) Neither are conserved

Example 6.5c

A proton scatters off another proton. No new particles are created.

A) Net momentum of two protons is conservedB) Net kinetic energy of two protons is conservedC) Both kinetic energy and momentum are conservedD) Neither are conserved

Perfectly Inelastic collision in 1-dimension

• Final velocities are the same

€

m1v1i + m2v2i = m1 + m2( )v f

Example 6.6

A 5879-lb (2665 kg) Cadillac Escalade going 35 mph =smashes into a 2342-lb (1061 kg) Honda Civic also moving at 35 mph=15.64 m/s in the opposite direction.The cars collide and stick.a) What is the final velocity of the two vehicles?

b) What are the equivalent “brick-wall” speeds for each vehicle?

a) 6.73 m/s = 15.1 mphb) 19.9 mph for Cadillac, 50.1 mph for Civic

Example 6.7

A proton (mp=1.67x10-27 kg) elastically collides with a target proton which then moves straight forward. If the initial velocity of the projectile proton is 3.0x106 m/s, and the target proton bounces forward, what are

a) the final velocity of the projectile proton?b) the final velocity of the target proton?0.03.0x106 m/s

Elastic collision in 1-dimension

1. Conservation of Energy:

2. Conservation of Momentum:

• Rearrange both equations and divide:€

12 m1v1i

2 + 12 m2v2i

2 = 12 m1v1 f

2 + 12 m2v2 f

2 (1)

€

m1v1i + m2v2i = m1v1 f + m2v2 f (2)

€

m1 v1i2 − v1 f

2( ) = m2 v2 f

2 − v2i2

( ) (1)

m1 v1i − v1 f( ) v1i + v1 f( ) = m2 v2 f − v2i( ) v2 f + v2i( )

m1 v1i − v1 f( ) = m2 v2 f − v2i( ) (2)

€

v1i + v1 f = v2 f + v2i

⇒ v1i − v2i = − v1 f − v2 f( )

Elastic collision in 1-dimension

Final equations for head-on elastic collision:

• Relative velocity changes sign

• Equivalent to Conservation of Energy€

m1v1i + m2v2i = m1v1 f + m2v2 f

v1i − v2i = − v1 f − v2 f( )

Example 6.8

An proton (mp=1.67x10-27 kg) elastically collides with a target deuteron (mD=2mp) which then moves straight forward. If the initial velocity of the projectile proton is 3.0x106 m/s, and the target deuteron bounces forward, what are

a) the final velocity of the projectile proton?b) the final velocity of the target deuteron?vp =-1.0x106 m/s

vd = 2.0x106 m/s

Head-on collisions with heavier objects always lead to reflections

Example 6.9a

The mass M1 enters from the left with velocity v0 and

strikes the mass M2=M1 which is initially at

rest. The collision is perfectly elastic. a) Just after the collision v2 ______ v0.

A) >B) <C) =

Example 6.9b

The mass M1 enters from the left with velocity v0 and

strikes the mass M2=M1 which is initially at

rest. The collision is perfectly elastic.

Just after the collision v1 ______ 0.

A) >B) <C) =

Example 6.9c

The mass M1 enters from the left with velocity v0 and

strikes the mass M2=M1 which is initially at

rest. The collision is perfectly elastic.

A) >B) <C) =

Just after the collision P2 ______ M1v0.

Example 6.9d

The mass M1 enters from the left with velocity v0 and

strikes the mass M2=M1 which is initially at

rest. The collision is perfectly elastic.

At maximum compression, the energy stored in the spring is ________ (1/2)M1v0

2

A) >B) <C) =

Example 6.9e

The mass M1 enters from the left with velocity v0 and

strikes the mass M2<M1 which is initially at

rest. The collision is perfectly elastic.

Just after the collision v2 ______ v0.

A) >B) <C) =

Example 6.9f

The mass M1 enters from the left with velocity v0 and

strikes the mass M2<M1 which is initially at

rest. The collision is perfectly elastic.

Just after the collision v1 ______ 0.

A) >B) <C) =

Example 6.9g

The mass M1 enters from the left with velocity v0 and

strikes the mass M2<M1 which is initially at

rest. The collision is perfectly elastic.

Just after the collision P2 ______ M1v0.

A) >B) <C) =

Example 6.9h

The mass M1 enters from the left with velocity v0 and

strikes the mass M2<M1 which is initially at

rest. The collision is perfectly elastic.

A) >B) <C) =

At maximum compression, the energy stored in the spring is ________ (1/2)M1v0

2

Example 6.9i

The mass M1 enters from the left with velocity v0 and

strikes the mass M2>M1 which is initially at

rest. The collision is perfectly elastic.

Just after the collision v2 ______ v0.

A) >B) <C) =

Example 6.9j

The mass M1 enters from the left with velocity v0 and

strikes the mass M2>M1 which is initially at

rest. The collision is perfectly elastic.

Just after the collision v1 ______ 0.

A) >B) <C) =

Example 6.9k

The mass M1 enters from the left with velocity v0 and

strikes the mass M2>M1 which is initially at

rest. The collision is perfectly elastic.

Just after the collision P2 ______ M1v0.

A) >B) <C) =

Example 6.9l

The mass M1 enters from the left with velocity v0 and

strikes the mass M2>M1 which is initially at

rest. The collision is perfectly elastic.

At maximum compression, the energy stored in the spring is ________ (1/2)M1v0

2

A) >B) <C) =