Question

A 77.49 -kg fisherman is sitting in his 28.31 -kg fishing boat along with his \(14.27-\mathrm{kg}\) tackle box. The boat and its cargo are at rest near a dock. He throws the tackle box toward the dock, and he and his boat recoil with a speed of \(0.3516 \mathrm{~m} / \mathrm{s}\). With what speed, as seen from the dock, did the fishermen throw his tackle box?

Short answer

Answer: The speed of the thrown tackle box, as seen from the dock, is 2.6066 m/s.

Step-by-step solution

Identify the initial and final momentum

Initially, the fisherman, the boat, and the tackle box are all at rest. So, the initial momentum of the entire system is 0. When the fisherman throws the tackle box, he and his boat will be recoiling with a speed of \(0.3516 \mathrm{~m} / \mathrm{s}\). Let's denote this speed as \(v_{recoil}\).

Calculate the total mass of the recoiling system

To calculate the total mass of the recoiling system, we need to add the mass of the fisherman and the fishing boat. Mass of fisherman (m1) = 77.49 kg, and mass of the fishing boat (m2) = 28.31 kg. So, the total mass of the recoiling system (M) is: \(M = m1 + m2 = 77.49 + 28.31 = 105.8 \mathrm{~kg}\).

Calculate the final momentum of the recoiling system

The final momentum of the recoiling system can be calculated using the mass of the recoiling system (M) and its recoil speed (v_recoil): \(momentum_{recoil} = M \times v_{recoil} = 105.8 \times 0.3516 = 37.20488 \mathrm{~kg~m}/\mathrm{s}\).

Calculate the final momentum of the tackle box

Since the initial momentum of the system is equal to the final momentum of the system, the final momentum of the tackle box will be equal to the negative of the final momentum of the recoiling system (fisherman and boat): \(momentum_{tacklebox} = -momentum_{recoil} = -37.20488 \mathrm{~kg~m}/\mathrm{s}\).

Calculate the speed of the thrown tackle box

To find the speed of the thrown tackle box (v_tacklebox), we will use its final momentum (momentum_tacklebox) and its mass (m_tacklebox = 14.27 kg): \(v_{tacklebox} = \frac{momentum_{tacklebox}}{m_{tacklebox}} = \frac{-37.20488}{14.27} = -2.6066 \mathrm{~m}/\mathrm{s}\). The negative sign indicates that the tackle box was thrown in the opposite direction of the recoiling system. So the speed of the thrown tackle box, as seen from the dock, is \(2.6066 \mathrm{~m}/\mathrm{s}\).

Key concepts

Momentum Conservation

Understanding the principle of momentum conservation is crucial when solving problems in physics, especially those involving collisions or separations. Momentum, a measure of the motion of an object, is calculated by multiplying the object's mass by its velocity. In the absence of external forces, the total momentum of a closed system must remain constant before and after any event. This is known as the Law of Conservation of Momentum.

In the context of the fisherman and his tackle box, this law implies that the total momentum before the fisherman throws the tackle box is equal to the total momentum afterward. Initially, they are all at rest, meaning their collective momentum is zero. Once the tackle box is thrown, the system—fisherman, boat, and tackle box—must still have zero total momentum. The momentum of the fisherman and boat moving in one direction is therefore balanced by the momentum of the tackle box moving in the opposite direction.

Recoil Velocity

When an object is propelled one way, a second object or a group of objects will move in the opposite direction; this is known as the recoil. The speed at which this second object or group moves is called the recoil velocity. This occurs due to the momentum conservation principle. The phenomenon is commonly seen in everyday activities such as a person on a skateboard throwing a ball forward and moving backward as a result.

In the exercise, after the fisherman throws his tackle box, he and his boat move backward with a certain velocity known as the recoil velocity. Calculating this recoil velocity correctly is essential to determine the speed at which the tackle box was thrown, considering the entire system was initially at rest.

Initial and Final Momentum

The initial momentum of a system is defined as the total momentum before an interaction takes place. On the other hand, the final momentum is the total momentum experienced by the system after the interaction. According to the conservation principle, these two should be equal if no external forces act on the system.

In our exercise, the initial momentum is zero because the fisherman, boat, and tackle box are at rest. The fisherman then throws the tackle box, leading to a recoil motion with corresponding velocities. The total momentum of the system (fisherman and boat combined, and the tackle box) after the tackle box is thrown must equal the initial momentum (zero in this case), taking into account the direction of the momentum.

System Mass Calculation

To apply the principles of momentum conservation, one needs to precisely calculate the mass of the associated objects or systems. The mass of a system is simply the sum of the masses of all the individual components. This calculation is vital as it directly affects the momentum calculation, where momentum equals mass multiplied by velocity.

In our scenario, the system mass calculation involves determining the sum of the mass of the fisherman and his boat to find the mass of the recoiling system. This mass is then used, along with the recoil velocity, to find the total momentum of the recoiling system after the tackle box is thrown.