Ms. Muse shoots a 0.005 toy dart at a toy boat with a plastic sail. The dart sticks to the sail of the boat and begins to move in the opposite direction. What is the final velocity of the boat if the boat was originally traveling toward Ms. Muse at 0.05 m/s, the mass of the boat is 0.42 kg, and the darts original velocity was 57.91 m/s?
0.005 *57.91 - 0.42 * 0.05 = 0.425 * v
To determine the final velocity of the boat, we can use the principle of conservation of momentum. The total momentum before the collision is equal to the total momentum after the collision. In this case, we need to consider the momentum of the boat and the momentum of the dart.
The momentum of an object can be calculated by multiplying its mass by its velocity. Let's denote the final velocity of the boat as Vf.
Before the collision:
Momentum of the boat = Mass of the boat * Initial velocity of the boat
= 0.42 kg * 0.05 m/s
Momentum of the dart = Mass of the dart * Initial velocity of the dart
= 0.005 kg * 57.91 m/s.
After the collision:
Momentum of the boat = Mass of the boat * Final velocity of the boat
= 0.42 kg * Vf
The total momentum before the collision is equal to the total momentum after the collision:
(Momentum of the boat before) + (Momentum of the dart before) = (Momentum of the boat after)
(0.42 kg * 0.05 m/s) + (0.005 kg * 57.91 m/s) = 0.42 kg * Vf
Now, we can solve for Vf:
(0.021 kg⋅m/s) + (0.28955 kg⋅m/s) = 0.42 kg * Vf
0.31055 kg⋅m/s = 0.42 kg * Vf
Divide both sides by 0.42 kg to isolate Vf:
Vf = 0.31055 kg⋅m/s / 0.42 kg
Vf = 0.7394 m/s
Therefore, the final velocity of the boat is approximately 0.7394 m/s.