A child in a boat throws a 5.15-kg package out horizontally with a speed of 10.0 m/s. The mass of the child is 24.2kg and the mass of the boat is 33.2kg . Calculate the velocity of the boat immediately after, assuming it was initially at rest.
use conservation of momentum.
masspackage*velocitypackage+totalmassinboart*velocityboat=0
solve for velocityboat.
To solve this problem, we need to apply the principle of conservation of linear momentum. According to this principle, the total momentum before an event is equal to the total momentum after the event.
Initially, the child, package, and boat are at rest, so the total momentum is zero. After the child throws the package, the total momentum remains zero, but now it is distributed between the child, the package, and the boat.
Let's denote the velocity of the boat after the throw as v_b, and the velocity of the child after the throw as v_c. The velocity of the package after the throw will be the same as its initial velocity, 10.0 m/s.
Using the principle of conservation of linear momentum, we can set up the following equation:
(m_child + m_package + m_boat) * 0 = m_child * v_c + m_package * 10.0 m/s + m_boat * v_b
Substituting the given values, we have:
(24.2 kg + 5.15 kg + 33.2 kg) * 0 = 24.2 kg * v_c + 5.15 kg * 10.0 m/s + 33.2 kg * v_b
0 = 24.2 kg * v_c + 51.5 kg m/s + 33.2 kg * v_b
Simplifying this equation, we get:
0 = 24.2 kg * v_c + 33.2 kg * v_b + 51.5 kg m/s
Now, we can solve for v_b:
v_b = -24.2 kg * v_c / 33.2 kg - 51.5 kg m/s
Since we are assuming the boat was initially at rest (v_b = 0), we can simplify the equation further:
0 = -24.2 kg * v_c / 33.2 kg - 51.5 kg m/s
Multiplying both sides by 33.2 kg:
0 = -24.2 kg * v_c - 33.2 kg * 51.5 kg m/s
Simplifying, we get:
0 = -24.2 kg * v_c - 1704.8 kg^2 m/s
Rearranging the equation:
24.2 kg * v_c = -1704.8 kg^2 m/s
Dividing both sides by 24.2 kg:
v_c = -1704.8 kg^2 m/s / 24.2 kg
v_c ≈ -70.50 m/s
So, the velocity of the child after throwing the package is approximately -70.50 m/s. Since the negative sign indicates the direction, we know that the child moves in the opposite direction to the throw.
Now, substituting this value back into the equation for v_b:
v_b = -24.2 kg * (-70.50 m/s) / 33.2 kg - 51.5 kg m/s
Simplifying this equation:
v_b ≈ -130.12 m/s
The velocity of the boat immediately after the throw is approximately -130.12 m/s. Again, the negative sign indicates the direction of motion.