Impulse and Momentum

ASTRONAUT Edward H. White II is in the zero gravity of space. By firing the gas-powered gun,

he gains momentum and maneuverability. Credit: NASA

Momentum DefinedMomentum is defined as the product of mass and velocity. (vector quantity) Units: kg m/s; g cm/s; slug ft/s

Momentum

m = 1000.0 kg 𝒑=(𝟏𝟎𝟎𝟎 .𝟎𝐤𝐠 ) (𝟏𝟔𝐦 /𝐬 )

𝒗=𝟏𝟔𝐦/𝐬𝒑=𝟏𝟔𝟎𝟎𝟎𝐤𝐠𝐦/𝐬

IMPULSE

Dt

Impulse:Impulse is a force acting for a small time interval Dt.

Example 1: The face of a golf club exerts an average force of 4000 N for 0.002 s. What is the impulse imparted to the ball?

Dt

Impulse:

The unit for impulse is the newton-second (N-s)

Other units are dyne-second and pound-second (lb-s)

In a ½ sheet of pad paper, discuss briefly the movie clip Spiderman 2 in the

application of impulse and momentum (their relationship):

Notice:

1. Train and its motion

2. How spiderman wants to stop the train using his legs, single shot of spiderweb

by each hand; and multiple shots of spiderwebs;

3. “stopper” in the railroad.

Impulse Changes Velocity and Momentum

Consider a mallet hitting a ball:

Impulse = Change in “mv”Impulse = Change in “mv”

Impulse and Momentum

Impulse = Change in momentum

Dt

F mv

A force F acting on a ball for a time Dt increases its momentum mv.

Conversion: 1.00 N.s = 1.00 kg m/s

Newton’s Second Law of Motion

Example 3: A 50-g golf ball leaves the face of the club at 20 m/s. If the club is in contact for 0.002 s, what average force acted on the ball?

Dt

F mv

Given: m = 0.05 kg; vo = 0;

Dt = 0.002 s; v = 20 m/s+

Choose right as positive.

Average Force:

0

Vector Nature of Momentum

Consider the change in momentum of a ball that is dropped onto a rigid

plate:

vo

vA 2-kg ball strikes the plate with a speed of 20 m/s and rebounds with a speed of 15 m/s. What is the change in momentum?

+

Dp = mv - mvo

Dp = 70 kg m/s

Dp = 70 kg m/s

Dp = (2 kg)(15 m/s) - (2 kg)(-20 m/s)

Impulse = Change in momentumImpulse = Change in momentum

Momentum

Impulse

More Sample Problems: 1. A 0.10-kg ball is thrown straight up into the air

with an initial sped of 15 m/s. Find (a) the ball’s initial momentum; (b) its momentum at its maximum height; and (c) impulse of the weight on the ball from initial to maximum height.

2. The force shown in the force vs. time diagram acts on a 1.5 kg object. Find (a) the impulse of the force, (b) the final velocity of the object if it is initially at rest.

0.0 1.0 2.0 3.0 4.0 5.00.0

1.0

2.0

3.0Fx (N)

t (s)

𝑨𝒓=𝒍𝒘𝑨𝒕=

𝟏𝟐𝒃𝒉

Momentum is conserved in this rocket launch. The

velocity of the rocket and its

payload is determined by the mass and velocity of the

expelled gases. Photo:

NASA

Conservation of Linear Momentum

Conservation of Momentum

The total momentum of an isolated system of bodies remains constant.

system = set of objects that interact with each other.

An isolated system is one in which the only force present are those between the objects of the system; that is there is no net external

force.

therefore

𝑭𝟐𝟏

𝑭𝟏𝟐

�⃑�21=𝐹 21 .∆ 𝑡=𝑚1𝑣1′ −𝑚1𝑣1

�⃑�12=𝐹12 .∆ 𝑡=𝑚2𝑣2′ −𝑚2𝑣2

∴ 𝑗21=− 𝑗12

𝑚1𝑣1′ −𝑚1𝑣1=− (𝑚2𝑣2

′ −𝑚2𝑣2 )

𝑚1𝑣1+𝑚2𝑣2=𝑚1𝑣1′ +𝑚2𝑣2

′

Since

Initial momenta = final momenta

Conservation of Momentum

𝑚𝑔𝑣𝑔 ′ 𝑚𝑟 𝑣𝑟+𝑚𝑔𝑣𝑔=𝑚𝑟𝑣𝑟′ +𝑚𝑔𝑣𝑔

′

𝑚𝑟 𝑣𝑟 ′Sample Illustrations:

Initially, the system is at rest:

Rocket:

Gas:

𝑣𝑔❑′=−

𝑚𝑟 𝑣𝑟 ′

𝑚𝑔

Conservation of Momentum Revisited: Into Details

1 2𝒎𝟏𝒗𝟏 𝒎𝟐𝒗𝟐

1 2𝒎𝟏𝒗𝟏 ′ ’

1 2𝑭𝟏𝟐

𝑭𝟐𝟏

𝑚1𝑣1+𝑚2𝑣2=𝑚1𝑣1′ +𝑚2𝑣2

′

APPLICATION: COLLISIONS and Conservation of Momentum

Simple Collisions (Head-on Collisions)

Example 01 A 730-N man stands in the middle of a frozen pond of radius 5.0 m. He is unable to get to the other side because of a lack of friction between his shoes and the ice. To overcome this difficulty, he throws his 1.2-kg physics book horizontally toward the north shore at a speed of 5.0 m/s. How long does it take him to reach the south shore?

A car with a mass of 9.00 x 102 kg is traveling at +15.0 m/s while an SUV with a mass of 1.80 x 103 kg is traveling at -15.0 m/s. (a) If the cars collide head-on, becoming entangled, find the velocity of the entangled cars after the collision. (b) If the SUV stops after collision and not entangled, what is the velocity of the car after collision?

Assignment:

1. What is the momentum of a 15-g sparrow flying with an initial speed speed of 12 m/s to the east? If the sparrow increases its speed to 14 m/s in the same direction, what is the impulse of its force?

2. Calculate the force exerted on a rocket, given that the propelling gases are expelled at a rate of 1000 kg/s with a speed of 60,000 m/s (at take off)

3. A 900-kg box car traveling at +20 m/s strikes a stationary second car. After collision, the box car stops and the second car travels with a velocity of +30 m/s. What is the mass of the second car?

Summary of Formulas:

Impulse = Change in momentumImpulse = Change in momentum

Momentum

Impulse

Conservation of MomentumConservation of Momentum

∆ �⃑�=𝟎 ;𝒎𝟏𝒗𝟏+𝒎𝟐𝒗𝟐=𝒎𝟏𝒗𝟏′ +𝒎𝟐

′