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Sketch the initial and final conditions Sketch the momentum diagram Known:
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mA vAi 575 kg 150 m/s mB vBi vAf 1575 kg 500 m/s
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vAi After (final) (mA mB)vf
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vf KE KEf KEi Fraction of KEi lost, KE/KEi
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a Use the conservation of momentum equation to find the final velocity pAi pBi pAf pBf mAvAi mBvBi (mA mB)vf vf
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(mAvAi mBvBi) (mA mB) (575 kg)(150 m/s) (575 kg
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Math Handbook
Isolating a Variable page 845
(1575 kg)(500 m/s) 1575 kg)
Substitute mA 575 kg, vAi 150 m/s, mB 1575 kg, vBi 500 m/s
767 m/s, in the direction of the motion before the collision b To determine the change in kinetic energy of the system, KEf and KEi are needed 1 KEf mv 2
2 1 (m mB)vf2 2 A 1 (575 kg + 1575 kg)(767 m/s)2 2
Substitute m Substitute mA mA mB 1575 kg, vf 767 m/s
575 kg, mB
632 104 J KEAi KEBi
1 1 m v 2 m v 2 2 A Ai 2 B Bi 1 (575 kg)(150 m/s)2 2
Substitute KEAi
1 m v 2, KEBi 2 A Ai
1 m v 2 2 B Bi
1 (1575 kg)(500 m/s)2 2
844 J Solve for the change in kinetic energy of the system KE KEf KEi 632 104 J 844 104 J Substitute KEf 212 104 J
KE K Ei 3 212 104 J 844 104 J
Substitute mA 575 kg, mB 1575 kg, vAi 150 m/s, vBi 500 m/s
632 104 J, KEi
844 104 J
c Calculate the fraction of the original kinetic energy that is lost
Substitute KE 211 104 J, KEi 844 104 J
251% of the original kinetic energy in the system was lost
Evaluate the Answer
Are the units correct Velocity is measured in m/s; energy is measured in J Does the sign make sense Velocity is positive, consistent with the original velocities
Section 112 Conservation of Energy
19 An 800-g bullet is fired horizontally into a 900-kg block of wood on an air table and is embedded in it After the collision, the block and bullet slide along the frictionless surface together with a speed of 100 cm/s What was the initial speed of the bullet 20 A 073-kg magnetic target is suspended on a string A 0025-kg magnetic dart, shot horizontally, strikes the target head-on The dart and the target together, acting like a pendulum, swing 120 cm above the initial level before instantaneously coming to rest a Sketch the situation and choose a system b Decide what is conserved in each part and explain your decision c What was the initial velocity of the dart 21 A 910-kg hockey player is skating on ice at 550 m/s Another hockey player of equal mass, moving at 81 m/s in the same direction, hits him from behind They slide off together a What are the total energy and momentum in the system before the collision b What is the velocity of the two hockey players after the collision c How much energy was lost in the collision
In collisions, you can see how momentum and energy are really very different Momentum is almost always conserved in a collision Energy is conserved only in elastic collisions Momentum is what makes objects stop A 100-kg object moving at 500 m/s will stop a 200-kg object moving at 250 m/s if they have a head-on collision However, in this case, the smaller object has much more kinetic energy The kinetic energy of the smaller object is KE 1 (100 kg)(50 m/s)2 125 J The kinetic energy of 2 1 the larger object is KE (200 kg)(250 m/s)2 625 J Based on the 2 work-energy theorem, you can conclude that it takes more work to make the 100-kg object move at 500 m/s than it does to move the 200-kg object at 250 m/s It sometimes is said that in automobile collisions, the momentum stops the cars but it is the energy in the collision that causes the damage It also is possible to have a collision in which nothing collides If two lab carts sit motionless on a table, connected by a compressed spring, their total momentum is zero If the spring is released, the carts will be forced to move away from each other The potential energy of the spring will be transformed into the kinetic energy of the carts The carts will still move away from each other so that their total momentum is zero
A bullet of mass m, moving at speed v1, goes through a motionless wooden block and exits with speed v2 After the collision, the block, which v 1 has mass mB, is moving 1 What is the final speed, vB, of the block 2 How much energy was lost to the bullet 3 How much energy was lost to friction inside the block
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