A person stands in a stationary canoe and throws a 5.16-kg stone with a velocity of 7.75 m/s at an angle of 31.5° above the horizontal. The person and canoe have a combined mass of 105kg. Ignoring air resistance and effects of the water, find the horizontal recoil velocity (magnitude and direction) of the canoe.
To find the horizontal recoil velocity of the canoe, we can apply the principle of conservation of momentum. According to this principle, the initial momentum of the system (person, canoe, and stone) should be equal to the final momentum of the system.
1. Calculate the initial momentum:
The initial momentum of the system is the sum of the momentum of the person and canoe (m1v1) plus the momentum of the stone (m2v2). Velocity is a vector quantity, so we need to consider both magnitude and direction.
m1 = mass of the person and canoe = 105 kg
v1 = velocity of the person and canoe = 0 m/s (since the canoe is stationary)
m2 = mass of the stone = 5.16 kg
v2 = velocity of the stone = 7.75 m/s (at an angle of 31.5° above the horizontal)
To calculate the horizontal component of the stone's velocity (v2x), we can use the formula v2x = v2 * cos(theta), where theta is the angle of the stone's velocity above the horizontal.
v2x = 7.75 m/s * cos(31.5°) ≈ 6.691 m/s
So, the initial momentum (before the stone is thrown) is:
initial momentum = (105 kg * 0 m/s) + (5.16 kg * 6.691 m/s) = 34.483 kg*m/s
2. Calculate the final momentum:
The final momentum of the system will be the momentum of the person and canoe after the stone is thrown. Since the stone has been thrown horizontally, with no vertical velocity, the final momentum in the vertical direction is zero.
Since momentum is a vector quantity, the momentum of the person and canoe after the throw will be in the opposite direction of the stone's momentum. Hence, the final momentum will have the same magnitude as the initial momentum but in the opposite direction.
final momentum = -34.483 kg*m/s
3. Calculate the recoil velocity of the canoe:
Since the initial and final momentum should be equal, we can equate them to find the velocity of the canoe after the stone is thrown.
final momentum = initial momentum
(m1 + m2) * vf = (m1 * v1) + (m2 * v2x)
Plugging in the values:
(105 kg + 5.16 kg) * vf = (105 kg * 0 m/s) + (5.16 kg * 6.691 m/s)
vf ≈ (5.16 kg * 6.691 m/s) / (105 kg + 5.16 kg)
vf ≈ 0.340 m/s
So, the magnitude of the horizontal recoil velocity of the canoe is approximately 0.340 m/s. The direction will be opposite to the direction of the stone's velocity.
Therefore, the horizontal recoil velocity of the canoe is approximately 0.340 m/s in the direction opposite to the stone's velocity.