Conservation of Momentum

Transcription

1 Conseration of Moentu We left off last with the idea that when one object () exerts an ipulse onto another (), exerts an equal and opposite ipulse onto. This happens in the case of a classic collision, such as one billiard ball striking another, and also happens in the not so obious exaple of a fire hose ejecting large aounts of water (why do you think the fire hose jups around so wildly) and the case of a rocket ship flying into space (the rocket is ejecting large quantities of fuel, exerting an ipulse onto that ass, and the fuel exerts an equal and opposite ipulse onto the rocket, causing the rocket ship to experience a change in its oentu and fly upward into the atosphere). So let s deelop this atheatically for and experiencing a collision: J on = F t on = the ipulse that exerts on J on = F t on = the ipulse that exerts on What do we know about the force that exerts on as it copares to the force that exerts on during the collision? How does the tie of ipact that experiences during the collision copare to the tie of ipact that experiences during the collision?

2 The force that exerts on is equal and opposite to the force that exerts on this, of course, is due to Newton s 3 rd Law and the tie of ipact is the sae since both objects are inoled in the ipact. So we can say, again: J on = - J on or equialently F t on = - F t on Now that we e redeeloped this point, lets restate what this tells us about the other side of the ipulse/oentu equation: First, we know: F t = on and F t = on -- this is the ipulse/oentu equation. So, if F t on = - F t on what can we say about the change in oentu of as it copares to the change in oentu of? = - or equialently p = - p This translates to the idea that during a collision the change in oentu of will equal the change in oentu of body.

3 We re now getting into this idea of the conseration of oentu. If during a collision, the change in oentu of one object is equal to the change in oentu of the other, then within the syste represented by those two bodies, no oentu has been lost and none has been gained, the total oentu of the syste has been consered. This law of physics is best illustrated by Newton s Cradle. Giant Newton's Cradle.p4 = - or p = - p does offer atheatical proof that oentu is consered in a collision, howeer, in the aboe for it is not so handy when trying to sole conseration of oentu probles. It just takes a little bit of algebraic anipulation and we ll hae it in a uch handier for.

4 We start off with the expression preiously deried and we expand the s. What do you get? Δ = Δ ( ) = ( ) f i f i Don t forget that the negatie sign on the right side of the aboe equation is there due to the fact that the forces and hence the ipulses are equal and opposite. The next step would be to distribute the ass. What do you get? f i = f i For the final step, siply get all the initials on the left side and the finals on the right side. What do you get? (if all your ters are negatie, just get rid of the negatie sign) i i = f f

5 i i = f f Moentu of before the collision Moentu of after the collision Moentu of before the collision Moentu of after the collision s we see here, the total oentu of the syste (coposed of and ) before the collision will be equal to the total oentu of the syste after the collision. This is the Law of the Conseration of Moentu.

6 The Law of Conseration of Moentu p before = p after This coers any situation where the oentu of soe object has been changed due to an ipulse exerted by soe other object. Saple Proble: fter being struck by a bowling ball, a 1.5 kg bowling pin sliding to the right at 3.0 /s collides head on with another 1.5 kg bowling pin initially at rest. Find the elocity of the second pin, if the first pin (after the collision) oes to the right with a elocity of.5 /s.

7 Conseration of Moentu Probles There are three basic types of conseration of oentu probles. For each type of oentu proble, the conseration of oentu equation takes a slightly different for. Type I: Objects experience a collision where following the collision they oe with different elocities. fter the collision the two objects oe separately. This is referred to as an elastic collision. Exaples: two cars crashing into each other and bouncing off one another; the cue ball hitting the side of the 9 ball causing it to oe off into the corner pocket. It should be clear that you would use the standard for of the conseration of oentu equation for Type I probles: i i = f f Saple Proble: 25.0 kg buper car oing to the right at 5.0 /s oertakes and collides elastically with a 35.0 kg buper car oing to the right. fter the collision, the 25.0 kg buper car slows to 1.5 /s to the right, and the 35.0 kg car oes at 4.5 /s to the right. Find the elocity of the 35.0 kg buper car before the collision.

8 Type II: Objects experience a collision and following the collision they oe off at the sae elocity as one body. The objects becoe essentially one object after the collision. This is referred to as an inelastic collision. Exaples: getting tackled by a linebacker in a gae of football; a bullet fired fro a gun becoing ebedded in a block of wood (the bullet-block oe off as one); standing on roller skates and catching your excited puppy as she jups into your ars; trains coupling. What for does the conseration of oentu equation take for these types of probles? i i = M f Saple Proble: 1500 kg car traeling at 15.0 /s to the south collides with a 4500 kg truck that is initially at rest at a stoplight. The car and truck stick together and oe together after the collision. What is the final elocity of the two-ehicle ass?

9 Type III: This is not your typical collision. n object begins as one ass and ends up as two asses with equal and opposite oentus. Exaples: a rifle shooting a bullet; an object exploding into two pieces; an astronaut throwing an object in space. What for does the conseration of oentu equation take for these types of probles? M i = f f In all the cases you ll confront at this leel, the ass will always initially be at rest. Gien this inforation, what happens to the aboe equation? 0 = f f or f = - f Saple Proble: During a spacewalk, an 85 kg astronaut becoes untethered fro the spaceship. She s drifted away fro the ship and the only way to get back is to throw a wrench directly away fro the ship. If the wrench has a ass of 1 kg, and the astronaut throws the wrench with a elocity of 20 /s, with what elocity will the astronaut be propelled back to the ship?