An apple of mass mA=0.25 kg is hanging with 1.5 m long string tied to a branch o

Question

An apple of mass mA=0.25 kg is hanging with 1.5 m long string tied to a branch of a tree. A dart gun fired at the apple penetrates into it, and the apple and the dart together rise up in the air from its initial position such that the string makes a 20o angle with the vertical. The mass of dart is md=20 grams;

1) Sketch a clear diagram

2) Calculate the speed of the dart.

I have tried this problem for several hours but haven't got anywhere. I am willing to give MAX karma points to anyone who can provide a thorough explanation and solutions.

Thanks.

Explanation / Answer

I can't draw you a diagram, but the speed of the dart that I come up with is 17.98 m/s.

We cannot use conservation of energy to look at the collision, since it is inelastic, but we can use conservation of energy once the dart has come to rest in the apple, derive the velocity of the dart+apple combination, then work backwards to get momentum.

First, the height: When a 1.5 m string moves 20 degrees off the vertical, it is now 1.5 * cos(20) meters from the top pivot down to the level of the apple measured directly down; therefore the apple has moved 1.5 m - 1.5 * cos(20) = 0.09046 meters higher off the ground.

mgh = (1/2) mv2 where the gravitational potential energy it has acquired by moving up 0.09046 meters must be equal to the kinetic energy it had when it began to swing higher. I get a velocity of 1.3322 m/s.

That means the momentum must have been (0.27 kg)(1.3322 m/s) = 0.3597 kg-m/s   where 0.27 kg is the mass of the apple+dart.

That momentum all had to come from the dart, which weighs 0.02 kg, therefore its speed had to be 17.98 m/s.

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