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In the collision pictured in [link] , two carts collide inelastically. Cart 1 (denoted carries a spring which is initially compressed. During the collision, the spring releases its potential energy and converts it to internal kinetic energy. The mass of cart 1 and the spring is 0.350 kg, and the cart and the spring together have an initial velocity of . Cart 2 (denoted in [link] ) has a mass of 0.500 kg and an initial velocity of . After the collision, cart 1 is observed to recoil with a velocity of . (a) What is the final velocity of cart 2? (b) How much energy was released by the spring (assuming all of it was converted into internal kinetic energy)?
Strategy
We can use conservation of momentum to find the final velocity of cart 2, because (the track is frictionless and the force of the spring is internal). Once this velocity is determined, we can compare the internal kinetic energy before and after the collision to see how much energy was released by the spring.
Solution for (a)
As before, the equation for conservation of momentum in a two-object system is
The only unknown in this equation is . Solving for and substituting known values into the previous equation yields
Solution for (b)
The internal kinetic energy before the collision is
After the collision, the internal kinetic energy is
The change in internal kinetic energy is thus
Discussion
The final velocity of cart 2 is large and positive, meaning that it is moving to the right after the collision. The internal kinetic energy in this collision increases by 5.46 J. That energy was released by the spring.
Mass A (2.0 kg) has an initial velocity of 4 m/s in the + x -direction. Mass B (2.0 kg) has an initial velocity of 5 m/s in the – x -direction. If the two masses have an elastic collision, what will be the final velocities of the masses after the collision?
(d)
Mass A has an initial velocity of 22 m/s in the + x -direction. Mass B is three times more massive than mass A and has an initial velocity of 22 m/s in the – x -direction. If the two masses have an elastic collision, what will be the final velocities of the masses after the collision?
Mass A (2.0 kg) is moving with an initial velocity of 15 m/s in the + x -direction, and it collides with mass B (5.0 kg), initially at rest. After the collision, the two objects stick together and move as one. What is the change in kinetic energy of the system as a result of the collision?
(c). Because of conservation of momentum, the final velocity of the combined mass must be 4.286 m/s. The initial kinetic energy is . The final kinetic energy is , so the difference is −161 J.
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