A girl is skipping stones across a lake. One of the stones accidentally ricochets off a toy boat that is initially at rest in the water (see the drawing below). The 0.058-kg stone strikes the boat at a velocity of 15 m/s, 15° below due east, and ricochets off at a velocity of 8 m/s, 12° above due east. After being struck by the stone, the boat's velocity is 2.2 m/s, due east. What is the mass of the boat? Assume the water offers no resistance to the boat's motion.
To find the mass of the boat, we can use the principle of conservation of momentum. According to this principle, the total momentum before the collision should be equal to the total momentum after the collision.
The momentum of an object is defined as the product of its mass and velocity. Mathematically, momentum (p) = mass (m) * velocity (v).
Before the collision:
The girl throws the stone, so the momentum of the stone is 0.058 kg * 15 m/s (magnitude of velocity).
The boat is initially at rest, so its momentum is zero.
After the collision:
The stone ricochets off the boat, so its momentum after the collision is 0.058 kg * 8 m/s (magnitude of velocity).
The boat's velocity after the collision is given as 2.2 m/s (magnitude of velocity).
To find the mass of the boat, we can set up the equation using the conservation of momentum:
Momentum before collision = Momentum after collision
(0.058 kg * 15 m/s) + (0 kg * 0 m/s) = (0.058 kg * 8 m/s) + (M * 2.2 m/s)
Simplifying and solving for M (mass of the boat):
0.87 kg m/s = 0.464 kg m/s + 2.2M
Subtracting 0.464 kg m/s from both sides:
0.87 kg m/s - 0.464 kg m/s = 2.2M
0.406 kg m/s = 2.2M
Dividing both sides by 2.2:
0.406 kg m/s / 2.2 = M
M ≈ 0.185 kg
Therefore, the mass of the boat is approximately 0.185 kg.