A child in a boat throws a 5.30 kg package out horizontally with a speed of 10 m/s, see the figure. Calculate the velocity of the boat immediately after, assuming it was initially at rest. The mass of the child is 26.0 kg and that of the boat is 55.0 kg
What is the magnitude?
Total linear momentum remains zero.
5.30 * 10 = -(26 + 55)*Vboat
Vboat = -0.6543 m/s
(opposite direction from thrown object)
Drop the minus sign for the magnitude.
It worked! You're awesome, thanks!
Well, let me try to calculate it with a twist of humor.
To find the magnitude of the velocity of the boat, we can use the principle of conservation of momentum. The initial momentum of the child and package is equal to the final momentum of the boat.
So, if we calculate the initial momentum of the child and package, we have:
Initial momentum of child and package = mass of child and package * velocity = (26.0 kg + 5.30 kg) * 10 m/s
Now, as the boat was initially at rest, its initial momentum is zero. Therefore, the final momentum of the boat is equal to the initial momentum of the child and package.
Final momentum of boat = initial momentum of child and package
Momentum equation:
(55.0 kg + 0 kg) * v = (26.0 kg + 5.30 kg) * 10 m/s
Simplifying the equation, we find:
55.0v = 31.3 * 10
Now divide both sides by 55.0:
v ≈ 5.69 m/s
So, the magnitude of the velocity of the boat immediately after is approximately 5.69 m/s.
To determine the magnitude of the velocity of the boat immediately after the package is thrown, we can use the principle of conservation of momentum.
The momentum of an object is given by the product of its mass and velocity. In this case, we have:
Initial momentum of the system = Final momentum of the system
The initial momentum of the system is zero since both the child and the boat are initially at rest. Therefore, we can express the momentum equation as:
(0 kg)(0 m/s) = (26.0 kg + 55.0 kg + 5.30 kg)(v)
Simplifying the equation:
0 = 86.3 kg (v)
To find the velocity (v) of the boat, we divide both sides of the equation by 86.3 kg:
v = 0 m/s
Therefore, the magnitude of the velocity of the boat immediately after the package is thrown is 0 m/s. This means that the boat does not move.
To calculate the magnitude of the velocity of the boat immediately after the package is thrown, we need to apply the principle of conservation of momentum.
The principle of conservation of momentum states that the total momentum of an isolated system remains constant, provided no external forces act on it. In this case, the child, package, and boat are the isolated system.
The initial momentum of the system is zero because both the child and the boat are initially at rest. The final momentum of the system after the package is thrown can be calculated by adding the individual momenta of the child, package, and the boat.
First, let's calculate the momentum of the child. The formula for momentum is given by:
Momentum = Mass * Velocity
Given:
Mass of the child (m1) = 26.0 kg
Velocity of the child (v1) = 0 m/s (because the child is initially at rest)
Momentum of the child (p1) = m1 * v1 = 26.0 kg * 0 m/s = 0 kg·m/s
Next, let's calculate the momentum of the package. The package is thrown horizontally with a speed of 10 m/s, so its vertical component of velocity is zero. Thus, only the horizontal component contributes to the momentum.
Given:
Mass of the package (m2) = 5.30 kg
Velocity of the package (v2) = 10 m/s
Momentum of the package (p2) = m2 * v2 = 5.30 kg * 10 m/s = 53 kg·m/s
Finally, let's calculate the momentum of the boat. Initially, the boat is at rest, so its velocity is zero.
Given:
Mass of the boat (m3) = 55.0 kg
Velocity of the boat (v3) = 0 m/s
Momentum of the boat (p3) = m3 * v3 = 55.0 kg * 0 m/s = 0 kg·m/s
Now, let's calculate the total momentum of the system after the package is thrown by adding the individual momenta:
Total momentum after = p1 + p2 + p3 = 0 kg·m/s + 53 kg·m/s + 0 kg·m/s = 53 kg·m/s
According to the principle of conservation of momentum, the total momentum of the system before and after the package is thrown should be equal. Since the initial momentum of the system was zero, the total momentum after the package is thrown should also be zero.
Therefore, the magnitude of the velocity of the boat immediately after the package is thrown is equal to the momentum of the system divided by the mass of the boat:
Magnitude of the velocity of the boat = Total momentum after / Mass of the boat
Magnitude of the velocity of the boat = 53 kg·m/s / 55.0 kg ≈ 0.96 m/s
So, the magnitude of the velocity of the boat immediately after the package is thrown is approximately 0.96 m/s.