Elastic and Inelastic Collisions
Explained
1) What is the significant difference, in terms of conservation of kinetic energy, between using elastic bumpers to cushion the collision and using a pin and soft plastic to couple the gliders? Describe each of the two types of collisions.
The following presumes that the duration of the events studied is so small that the effects of friction or other external forces are negligible.
A perfectly elastic bumper is not deformed by a collision. Therefore there is no heat loss or other energy loss associated with the collision. In a collision of two objects in which the objects bounce apart and there is no energy loss, kinetic energy and momentum both are conserved. That is, the total momentum of the system after the collision is the same as the total momentum of the system before the collision occurred. Also, the total kinetic energy of the system after the collision is the same as the total kinetic energy of the system before the collision occurred. This is called an elastic collision. When two objects collide and stick together, momentum is conserved, but kinetic energy is not conserved. This is called an inelastic collision. The kinetic energy of the system is decreased as it is transformed into other types of energy, such as heat when the soft plastic is deformed.
2) Is there a comparable difference, in terms of conservation of momentum, between the two types of collisions? What seems to be true about the effect of the collision on total momentum that differs from its effect on kinetic energy?
In elastic and inelastic collisions, there is no difference as far as conservation of momentum is concerned. The total momentum of the system is conserved in both types of collisions. The effect on kinetic energy is different. The total kinetic energy of the system is conserved in an elastic collision, but not in an inelastic collision.
A. Collisions
with Coupling
(See below for answers)
1. A girl, mass 70.0 kg, is running 3.0 m/s east when she jumps onto a stationary skateboard, mass 2.0 kg. What is the velocity of the girl and skateboard assuming they move off together?
2. A wrestler is standing at rest.. Another wrestler, running at 5.0 m/s, grabs him and holds onto him, and the two move off together with a velocity of 2.7 m/s in the same direction in which the second wrestler was running. If the mass of the second wrestler is 100 kg, what is the mass of the first wrestler?
3. A 1 kg glider and a 3 kg glider both slide toward each other at 1 m/s on an air track. They collide and stick. what will be the speed of the combined mass?
4. A bullet of mass, "m," is being fired with a velocity, "v," into a block of mass, "M." Find the loss in kinetic energy of the system.
5. A 6.00 g bullet is fired horizontally into a 1.50 kg wooden block resting on a horizontal surface. The coefficient of kinetic friction between block and surface is 0.30. The bullet remains embedded in the block, which is observed to slide 0.25 m along the surface before stopping. What was the initial speed of the bullet?
6. A bullet with mass 8.00 g is fired into a block with mass 0.914 kg. The block rests on a frictionless, horizontal surface and is attached to a coil spring. The spring 15.0 cm after the impact. The spring was found to compress 0.230 cm when a 0.750 N force was applied.
(a) Find the magnitude of the block's velocity just after impact.
(b) What was the initial speed of the bullet?
7.
a. Find the speed at which a super hero (mass=76.0 kg) must fly into a train (mass = 19537 kg) traveling at 65.0 km/hr to stop it.
b. Calculate the time the super hero must take to stop the train, if the passengers experience an average horizontal force of 0.550 their own weight.
8.
A boy weighing 140 pounds traveling at a speed of 15 mph collides headon with a boy weighing 130 pounds traveling at 12 mph. If the impact lasted 2 s, how much force is exerted?
B. Collisions without Coupling
(See below for answers)
1. A toy truck, with mass 20.0 g, travels along a level tabletop at 0.50 m/s. A miniature car, with mass 5.00 g, speeds headlong toward the toy truck at 0.75 m/s. Immediately after the collision, the toy truck continues in its original direction at 0.10 m/s. What is the velocity of the miniature car?
2. A 1kg ball moving at a speed of 10 m/s in the positive x direction collides head on with a 2 kg ball at rest. What are the velocities of the 1kg and 2 kg ball after the collision?
3. An 74.5 kg object moving to the right at 34.3 cm/sec overtakes and collides elastically with a second 55.8 kg object moving in the same direction at 16.7 cm/s. Find the velocity of the second object after the collision.
4. In an inelastic collision, is the final total momentum more,less, or the same as the initial momentum?
5. A 1700 kg car is moving 21 m/s [west], and a 3400 kg truck is traveling 14 m/s [east]. Find the velocity of the center of mass of the system.
6. A 0.4kg mass traveling 3 m/s hits a 0.6kg mass initially at rest. find The speed of the 0.4kg mass if the speed of the 0.6kg mass after the collision is 2.4 m/s. Is this collision an elastic collision or an inelastic collision?
7. The center of mass of a twoparticle system is located on the x axis at x = 5.0 m and has a velocity of 6.0 m/s. One of the particles is at the origin. The other particle has a mass of 0.30 kg and is at rest on the x axis at x = 8.0 m.
(a) What is the mass of the particle at the origin?
(b) Calculate the total momentum of this system.
(c) What is the velocity of the particle at the origin?
8. Consider elastic collisions between neutrons (1 amu) and deuterons (2 amu).
(a) What is the speed of a neutron, expressed as a fraction of its original speed, after a headon collision with a deuteron which is initially at rest?
(b) What is its kinetic energy, expressed as a fraction of its original kinetic energy?
(c) How many such successive collisions will reduce the speed of a neutron to 1/27 of its original value?
C. Explosions
(See below for answers)
1. A boy, mass 70.0 kg, riding a skateboard, mass 2.0 kg, is traveling 3.0 m/s east when he attempts to jump forward from his skateboard. If his velocity immediately after leaving the skateboard is 3.1 m/s [E], what is the velocity of t he skateboard?
2. A 232 Th (thorium) nucleus at rest decays to a 228 Ra (radon) nucleus with the emission of an alpha particle. The total kinetic energy of the decay fragments is 6.54 x 10^{3} J. An alpha particle has 1.76% of the mass of a 228 Ra nucleus.
a) Calculate the kinetic energy of the recoiling 228 Ra nucleus.
b) Calculate the kinetic energy of the alpha particle.
3. A child in a boat throws a 6.4 kg package out horizontally with a speed of 10 m/s. Calculate the velocity of the boat immediately after, assuming it was initially at rest. The mass of the child is 26 kg and that of the boat is 45 kg.
4. A 30.5 kg girl is standing on a 90 kg crate. The crate is at rest on a frictionless conveyer belt. When the girl walks on the crate at a constant 2.17 m/s to right relative to crate, what is her velocity relative to the conveyer belt?
5. A 13.8 kg block and a 25.3 kg block are resting on a horizontal frictionless surface. Between the two is a spring (spring constant = 1620 N/m). The spring is compressed by 0.143 m. With what speed does each block move when the system is released?
6. A 11.9 kg gun at rest contains a 0.0018 kg bullet. When fired, the bullet leaves the gun with a forward velocity of 290 m/s. What is the recoil velocity of the gun in m/s?
7. A 160 pound man jumps off an 800 pound raft, travels 15 ft horizontally, and lands in the water. What maximum distance separates the man and the raft when he hits the water?
D. Other
(See below for answers)
1. Two people, one of mass 72.8kg and the other of mass 52.4kg, sit in a rowboat of mass 81.6 kg. With the boat initially at rest, the two people, who have been sitting at opposite ends of the boat, 3.14m apart from each other, exchange seats. How far will the boat move?
ANSWERS
A1. mv_{1} + mv_{2} = m_{(1+2)}v_{(1+2)}
(70.0 kg)(3.0 m/s [E]) + (2.0 kg)(0 m/s) = (72.0 kg)v_{(1+2)}
v_{(1+2)} = 2.92 m/s [E]
For solutions to all the problems on this page click here.
B1.
Establishing the original direction of the toy truck as "+" and the original direction of the miniature car as therefore ""
m_{1}v_{1i} + m_{2}v_{2i} = m_{1}v_{1f} + m_{2}v_{2f}
(20.0 g)(0.50 m/s) + (5.00 g)(0.75 m/s) = (20.0 g)(0.10 m/s)+ (5.00 g)v_{2f}
v_{2f} = 0.85 m/s [in the direction of the toy truck before collision]
For solutions to all the problems on this page click here.
C1.
m_{(1+2)}v_{(1+2)} = mv_{1} + mv_{2}
(72.0 kg)(3.0 m/s [E]) = (70.0 kg)(3.1 m/s [E]) + (2.0 kg)v_{2}
v_{2} = 0.50 m/s [W]
For solutions to all the problems on this page click here.
