A child in a boat throws a 5.60 kg package out horizontally with a speed of 10.0 m/s. Calculate the velocity of the boat immediately after, assuming it was initially at rest. The mass of the child is 22.0 kg and that of the boat is 55.0 kg.
M1V1 = M2V2 so
5.6*10 = (22+55)V2
56/77 = V2
0.727
To solve this problem, we can use the principle of conservation of momentum. The total momentum before the package is thrown is equal to the total momentum after the package is thrown.
Let's define the positive direction as the forward direction of the boat.
First, let's find the initial velocity of the child and the boat together.
The total initial momentum before the package is thrown is:
Initial momentum = (mass of child + mass of boat) * initial velocity
Initial momentum = (22.0 kg + 55.0 kg) * 0 m/s (since the boat is initially at rest)
Initial momentum = 0 kg⋅m/s
Now, let's calculate the momentum of the package. Since it is thrown horizontally, there is no vertical component to consider. The momentum of the package is:
Momentum of the package = mass of the package * velocity of the package
Momentum of the package = 5.60 kg * 10.0 m/s
Momentum of the package = 56.0 kg⋅m/s
According to the principle of conservation of momentum, the total momentum after the package is thrown is equal to the initial momentum. Therefore,
Total momentum after = Total momentum before
(mass of child + mass of boat + mass of package) * final velocity = 0 kg⋅m/s
(22.0 kg + 55.0 kg + 5.60 kg) * final velocity = 0 kg⋅m/s
77.6 kg * final velocity = 0 kg⋅m/s
Now, we can solve for the final velocity of the boat by dividing both sides by 77.6 kg:
final velocity = 0 kg⋅m/s / 77.6 kg
final velocity = 0 m/s
Therefore, the velocity of the boat immediately after the package is thrown is 0 m/s.
To solve this problem, we can apply the principle of conservation of momentum. The total momentum of an isolated system remains constant before and after an event.
The initial momentum of the system, consisting of the child, the boat, and the package, is zero since the boat is initially at rest.
The final momentum of the system is the sum of the momentum of the boat and the child after the package is thrown.
Step 1: Calculate the momentum of the package.
Momentum (p) is calculated by multiplying the mass (m) by the velocity (v).
Momentum of the package = mass of the package × velocity of the package.
Given:
Mass of the package (m_package) = 5.60 kg
Velocity of the package (v_package) = 10.0 m/s
Step 2: Calculate the momentum of the child.
Given:
Mass of the child (m_child) = 22.0 kg
The child throws the package horizontally. Therefore, there is no vertical component of momentum.
Momentum of the child (p_child) = mass of the child × velocity of the child.
Step 3: Calculate the momentum of the boat.
Given:
Mass of the boat (m_boat) = 55.0 kg
Since the boat is initially at rest and there is no external force acting on it, the momentum of the boat before and after the event is zero.
Step 4: Apply the principle of conservation of momentum.
The total momentum before the event is zero, and after the event, it is the sum of the momentum of the child and the boat.
Total momentum before = Total momentum after
0 = p_child + p_boat
To simplify the calculation, we can write the equation as:
0 = (m_child × v_child) + (m_boat × v_boat)
Now we can solve for the velocity of the boat (v_boat).
Step 5: Rearrange the equation and solve for v_boat.
To find the velocity of the boat, we rearrange the equation:
-v_boat = (m_child × v_child) / m_boat
v_boat = -((m_child × v_child) / m_boat)
Now substitute the given values and calculate the result.
v_boat = -((22.0 kg × 10.0 m/s) / 55.0 kg)
v_boat = -4.00 m/s
Therefore, the velocity of the boat immediately after the package is thrown is -4.00 m/s. The negative sign indicates that the boat moves in the opposite direction of the thrown package.