a pirate ship fires a 25kg cannonball with a velocity of +100m/s at a very heavy fur trader ship. the ship is at rest. the cannonball bounces off the side of the ship, making a hole. after the collision the cannonball has a velocity of -0.5m/s. assume the water did not absorb any shock. how much energy was dissipated? what kind of collision is this? what is the coefficient of restitution?
To determine the dissipated energy, type of collision, and coefficient of restitution, we need to utilize the principles of conservation of momentum and energy.
1. First, let's calculate the initial and final momentum of the cannonball:
- Initial momentum (before collision): p1 = m1 * v1, where m1 is the mass of the cannonball (25 kg) and v1 is the initial velocity (+100 m/s).
- Final momentum (after collision): p2 = m1 * v2, where v2 is the final velocity (-0.5 m/s).
Plugging in the given values:
- p1 = (25 kg) * (+100 m/s) = 2500 kg·m/s
- p2 = (25 kg) * (-0.5 m/s) = -12.5 kg·m/s, considering negative direction.
2. Next, using the principle of conservation of momentum, we can determine the momentum of the fur trader ship:
- Since the fur trader ship is at rest (initial momentum = 0), the final momentum should also be zero, as the system is isolated.
- Therefore, the fur trader ship's momentum after the collision is zero: p_furship = 0 kg·m/s.
3. Now, we can calculate the change in momentum (Δp) of the cannonball during the collision:
- Δp = p2 - p1 = -12.5 kg·m/s - 2500 kg·m/s = -2512.5 kg·m/s.
4. The dissipated energy during the collision can be determined by using the equation:
- Energy = Change in kinetic energy = (1/2) * m1 * (v2^2 - v1^2), where m1 is the mass of the cannonball, v2 is the final velocity, and v1 is the initial velocity.
Plugging in the given values:
- Energy = (1/2) * (25 kg) * ( (-0.5 m/s)^2 - (+100 m/s)^2 )
= (1/2) * (25 kg) * (0.25 m^2/s^2 - 10,000 m^2/s^2)
= (1/2) * (25 kg) * (-9,999.75 m^2/s^2)
= -124,996.88 J (assuming positive energy is dissipated).
Therefore, the dissipated energy is approximately 124,996.88 Joules.
5. To determine the type of collision, we can observe the signs of the final velocities:
- The initial velocity of the cannonball is positive, while the final velocity is negative.
- Since the sign of the initial and final velocities are different, this indicates an elastic collision.
6. The coefficient of restitution (e) can be calculated using the formula:
- e = |v2| / |v1|, where v2 is the final velocity and v1 is the initial velocity.
Plugging in the given values:
- e = |-0.5 m/s| / |100 m/s|
= 0.005
Therefore, the coefficient of restitution is approximately 0.005.