Chapter 16 Momentum & Impulse While momentum has many common meanings it definition in physics is quite specific. One of the more amazing concepts in physics is the conservation of momentum. When you fire a shotgun at a clay target the shot going out the barrel is given a certain momentum. The recoil momentum of the shotgun is equal to the momentum of the shot.
The momentum of an object is calculated in a very easy way. where p is used to represent momentum, m is the mass and v is the velocity of the object. Notice that momentum is a vector so we need to know both the magnitude and direction for momentum. The typical units we will use for momentum are: pucks then letting go. You are having fun and attracting quite a crowd as the masses slide off in opposite directions. You decide to really have some fun by measuring one of the pucks after the "explosion" and trying to determine what the other puck must be doing. 1. You grab a 3 kg and a 7 kg puck and one of your springs. You squeeze the pucks together with the spring in between. You let the contraption go and the 3 kg mass slides across the floor to the North with a velocity of 10 m/s. How fast and in what direction will the 7 kg mass move? 2. More waiting, the plane is delayed, so you grab an 8 kg and a 20 kg puck and a new spring. After the spring flings the pucks apart, the 20 kg puck is traveling at 2.6 m/s to the East. How fast and in what direction is the 8 kg puck moving? Which we abbreviate as: Since momentum is a vector quantity we must use vectors in two and three dimensional collisions. You are waiting at La Guardia airport for your ship to come in. (huh). To pass the time you bring with you an assortment of pucks with rollers on them (to reduce friction) and a couple of springs. You begin by squishing the spring between the two 3. It is fun to be an astronaut. You are floating free along side your space capsule. The capsule is almost 30 m length and has a mass of 1,000 kg. It is somewhat difficult to move around, the lack of a local gravitational field is unusual. Also, your 40 kg space suit almost doubles your mass. With the space suit on you have a mass of 90 kg. As you float near your space capsule you notice that you are out of fuel. This is not a good thing, you think to yourself and you regret having taken off the tether rope which connected you to the capsule. You notice that you are about 3 m away from the side of the capsule and that you do not seem to be moving either towards or away from the capsule (i.e. 104
your velocity wrt the capsule is zero). You take a 0.40 kg wrench which is about 10 cm long out of your pocket and throw it at 18 m/s in a direction opposite to the direction back to the capsule. a) What will be your speed after the throw? b) How long will it take you to get back to the space capsule? 4. After seeing the incredible publicity that surrounded your near encounter with the abyss, Ned decides to do the same thing. He takes out his lucky penny and lets it go. He is always amazed that the penny just stays suspended a few centimeters above the capsule. He stands on the edge of the capsule and pushes off. He notices that he is moving with a velocity of 2.0 m/s away from the penny. He also notices to his chagrin that the capsule is moving in the opposite direction with respect to the penny. Ned wonders if it has something to do with the fact that his mass with space suit is 10 kg more than yours. a) How fast is the capsule moving away from the penny? b) How far will Ned be from the capsule after 10 s? c) Panicking, Ned takes out his 0.4 kg wrench and throws it with a speed of 30 m/s, as the clock hits the 15 s mark of his journey, in a direction opposite of where he needs to go to get back to the capsule. Will he make it back to the capsule? d) How far will Ned be away from the capsule at the 100 s mark (that is 100 s after he leaves the capsule)? Practicing for the Olympic small bore creamed corn can shooting competition, freshwoman Molly Monk, who stands 175 cm tall, fires a 22 g bullet from her Walther competition pistol. The pistol has a mass of 1.2 kg. The bullet is traveling with a speed of 620 m/s when it strikes a 0.411 kg creamed corn can. The bullet goes through the can (in medical parlance this is called a through and through) and exits the can with a velocity of 540 m/s. 5. How fast is the can moving? 6. You place a 2.5 kg can of cooked tomatoes on top of a fence post. The post is 1.4 m high and is about 100 m in front of a 30 m tall hill. You take 10 paces back, draw your pistol and fire at the cooked tomatoes. The 30 g bullet strikes the can with a velocity of 620 m/s and does a through and through. The bullet exits the can with a speed of 370 m/s. a) What is the speed of the can after the bullet leaves the can? Bored to tears because your physics class had been preempted today with another required assembly you begin to play around with a rubber ball. You really like the wood floors in your living room and keeping them nice and polished really reduces the effects of friction in your playful experiments. You roll your 30 g rubber ball with a velocity of 148 cm/s across the floor. The plot thickens, the speed of light remains the same. An unsuspecting 105
cucumber lies motionless. Earlier you had determined that the mass of the cucumber was 200 g. The ball strikes the cucumber. An amazing collision as the ball strikes the cucumber head on right on its center of mass. The ball recoils back towards you with a velocity of 80 cm/s. 7. How fast and in what direction is the cucumber moving? 8. A 5.7 kg rifle fires a 40 g bullet with a velocity of 640 m/s. What is the recoil speed of the rifle? 9. Softball hockey, another attempt at a unique combination sport. While gliding along the ice with a speed of 2.1 m/s Hilda catches a 0.8 kg ball traveling at 27 m/s (wrt the ice). Hilda was traveling directly towards the ball. After the catch Hilda notices that her speed has slowed to 1.2 m/s. Hilda is about 160 cm tall. a) What is Hilda's mass? b) How old is Hilda? 10. The Queen Mary, what a ship, 2.3 x 10 7 kg of floating palace. You decide to take a dive off the bow of the Queen Mary. You push off and notice that your horizontal velocity is 4.8 m/s with respect to the dock. a) What will be the recoil speed of the Queen Mary? b) How long will it take the Queen Mary to move 1 m at this speed (ignore friction)? 11. A 50 g bullet, traveling at 580 m/s, strikes a 9.17 kg block of wood sitting on a 1.0 m high oak table. The bullet embeds itself in the block. How fast does the block move after the collision? 12. A blob of clay, mass of 4 kg, scooting along the frictionless surface of the art room floor with a velocity of 8 m/s towards the front of the room, runs head on with a 3 kg block of clay traveling at 6 m/s towards the back of the room. The two blobs stick together. a) How fast will the two blobs be going after the collision? b) How many points will you get for doing this? 13. A 45 g bullet imbeds in a 2.1 kg block of wood, sending it flying at 10.7 m/s. What was the velocity of the bullet just before it struck the wood? 14. A 120 g Hostess Twinkie, sliding across the table at 270 cm/s strikes head on a stationary double Twinkie (m= 240 g). If the two Twinkies stick together, what will be the speed of the stuck together Twinkies? 15. (optional) A 2 kg can of beans, sitting on top of a 1.4 m tall fence post, is struck by a 50 g bullet traveling at 400 m/s. If the can falls to the ground 0.8 m away from the base of the fencepost, what is the exit speed of the bullet? (this requires that you remember your kinematics equations from the first couple of chapters) 106
(to be said in a circus announcer's voice) And now for your further edification and enjoyment we will take on the challenge of the two dimensional momentum conservation problems. So hold onto your hat (actually you can get a point for wearing a hat... so let your hat go), sit forward and remember your trig. <parent mode> sit up straight, are you sure you have enough light? do your homework etc. etc. </parent mode> These blobs of frictionless clay are really kind of fun to play with. (ending another sentence with a preposition oh well ). You pick up a 2 kg hunk of the clay and shape it into a whale, meanwhile your friend Mindy, takes a 1.5 kg piece and makes a likeness of Jabba the Hut. (OK so Mindy is a little strange). You push your whale and it slides across the floor heading towards the front (East) with a velocity of 2.0 m/s. Mindy pushes Jabba such that Jabba is traveling South (towards the windows) with a velocity of 1.7 m/s. They collide and stick together. 16. What will be the velocity (both direction and magnitude) of the WhaleJabba? 17. Mindy now picks up a 1.4 kg can of anchovies and slide the can across the floor with a speed of 5.7 m/s in a direction 30 North of East. You slide a flat rock, whose mass is about 1.0 kg, left over from the annual Rock Skipping Days at Lake Manawa and slide the rock due South with a speed of 3 m/s. The rock is blue green in color with a cylindrical shape with a diameter of about 10 cm.. The two objects smash into each other and surprisingly stick together. What is the speed and direction of the anchovies after the collision? 18. Mindy sends another can sliding across the floor. This time the can is full of dried cats and has a mass of 1.4 kg. The can is traveling due North at a speed of 5.7 m/s when it strikes a 0.55 kg can of Spam traveling due East at 8.8 m/s. The collision is elastic. The spam is seen traveling after the collision at a speed of 6.8 m/s with a direction 65 North of East. What is the speed and direction of the can of dried cats after the collision? 19. One last set of Mindy problems (at least for now)... You launch a 2 kg blob of frictionless clay to the East with a speed of 6 m/s. Mindy launches a 1.5 kg blob of clay pushing to the North. The clay blobs collide and stick together. They move off stuck together at an angle of 33 North of East. a) What is the magnitude of the velocity of the stuck together clay blobs? 107
b) What was the velocity of Mindy's clay just before it struck your blob? Impulse equals the change in momentum of an object and can be determined either by measuring the change in momentum or by knowing the net force involved and the duration of the force. Francie and Ned are at the rolling rink, trying a new maneuver. Francie, whose mass is 48 kg, hugs all 66 kg of Ned. They start at rest and push away from each other. After this Francie is rolling backward at 3.7 m/s. 22. How fast is Ned going? 23. You measure the force of a model rocket motor as it goes through its thrust phase. The force is not constant and a graph is provided below. You attach the rocket so that it is pointing out 20. A 3 kg banana (a big banana) speeds up from 4 m/s to 6 m/s the back of a 2.0 kg toy Ricky Rudd race car. The car has very when it is struck from behind by a squash. What impulse did the banana receive? (i.e what is the change in the banana's momentum?) Force vs time graph for model rocket motor. Ann wants to be taller. One way in which she explores this fantasy is by standing on chairs. Ann weighs about 400 N. She jumps off a chair and lands on the floor. Her velocity on impact is 1.7 m/s. Her knees are somewhat locked so she comes to a complete stop in 0.016 s. 21. What force in excess of her weight does Ann experience while stopping? (this is why you should bend your knees while landing. If you bend your knees you extend the time it takes to stop. This reduces the force of the collision. 108
little friction. The car is about 25 cm long and 12 cm wide. You fire the rocket. a) What impulse is given to the Ricky Rudd race car? b) How fast will the car be going when the engines burn out? c) How far will the race car go while the engines are running? d) How long will it take for the car to travel 30 m? You are at the bottom of a shallow well. Who knows how you got there. The last thing you remember is someone saying "rosebud" and another deeper voice saying "could be worse, could be raining" You hate it when movie quotes seem to appear at random in your brain. The top of the well is somewhat blocked by a 5.4 kg block of wood laying across the opening of the well. You can't reach the board. In a flash of brilliance, you take out your Walther target pistol, left over from the Olympics. You fire one 30 g bullet straight up. It strikes the board and the imbeds itself into the block. The board lifts into the air from the collision and the board/bullet combination rises 1.8 m before falling back down over the well. 25. You are hovering in deep space about 400 m from your space craft. As usual, you are not tethered. You arrange yourself so that when you fire your personal thrusters it will propel you back to the craft. The thrusters however, do not perform so well and the force they provide is quite variable as seen by the graph below. If you have a total mass of 100 kg when in your space suit. Force vs time graph for personal thrusters. 24. What was the speed of the bullet just before impact with the board? 109
a) How fast will you be going after your thruster burn? b) How far will you travel during the burn? c) How long will it take you to get back to your ship? 26. An 8 kg mass explodes into two parts, one traveling to the left at 11 m/s, the other to the right at 4 m/s. What are the masses of each part? A crow whose mass is 1.2 kg is flying over the Platte river with a speed of 15 m/s. He spots an oversized gnat hovering ahead. The gnat has a mass of 0.4 kg and is basically just hovering over one spot. The crow is hungry, open his mouth and grabs the gnat. 27. What will be the velocity of the crow immediately after nabbing the gnat? 27. Ned (mass = 70 kg) is hanging from a tree branch, overhanging the sidewalk you are skateboarding on. You are toodling along at 2.4 m/s when just as you pass the tree Ned lets go and lands on your skateboard. He puts his arms around you to steady himself, you look back at him, your eyes meet... yeeeech.. back to the physics. If your mass (with the skateboard) is 55 kg, what will be your velocity right after Ned lands on your skate board? Sitting by the railroad tracks near Broken Bow Nebraska, the only radio station you can tune in is a country western station. It is playing one of the country classics by Loretta Lynn titled "It don't do no good to be a good girl". You wonder about her grammar, when along comes a railroad car, its 6,000 kg mass traveling at 2.3 m/s. It is on the same track as the stationary car before you. The car in front of you only has a mass of 4,000 kg. The moving car couples with the stationary one and they go off together. 28. How fast will the two cars be going right after the collision? 29. A 40 g bullet slows from 600 to 500 m/s while penetrating a can of anchovies. The bullet passes through the center of the can. The diameter of the can is 12 cm, the mass of the can is about 1 kg. and the distance from Omaha to Denver is about 1000 km. a) What impulse did the can receive? b) How fast will the can be moving after the bullet exits the can? You are traveling at a high rate of speed down the interstate, well about 150 km/hr, when suddenly your car goes out of control and you smash head on into a concrete retaining wall. 30. If you are not wearing your seat belts you head will be stopped by the windshield. Assuming that the windshield does not break, and that it stops you in 0.015 s, how much force is being applied to you while stopping? 110
31. If you have an airbag (and it deploys) how does that lessen your chance for injury? (be specific with both an explanation and some numerical examples) Great people talk about ideas, average people talk about events, small people talk about people 111