Question

A 75.19 -kg fisherman is sitting in his 28.09 -kg fishing boat along with his \(13.63-\mathrm{kg}\) tackle box. The boat and its cargo are at rest near a dock. He throws the tackle box toward the dock with a speed of \(2.911 \mathrm{~m} / \mathrm{s}\) relative to the dock. What is the recoil speed of the fisherman and his boat?

Short answer

Answer: The recoil speed of the fisherman and his boat is 0.384 m/s.

Step-by-step solution

Calculate the Initial Momentum

Initially, the fisherman, boat, and tackle box are all at rest. So, the initial total momentum is 0.

Set up the Conservation of Momentum Equation

Let v be the recoil speed of the fisherman and his boat. (initial momentum) = (final momentum) 0 = (m1 + m2) * v + m3 * v1

Plug in the Values

Now, we will plug in the values of the given data into the conservation of momentum equation. 0 = (75.19 kg + 28.09 kg) * v + (13.63 kg)(2.911 m/s)

Solve for the Recoil Speed v

Rearrange the equation to solve for v: v = -(13.63 kg)(2.911 m/s) / (75.19 kg + 28.09 kg) Now, do the arithmetic: v = -(39.66 kg m/s) / (103.28 kg) v = -0.384 m/s

Write the Final Answer

Since we got a negative value for v, this indicates that the recoil speed of the fisherman and his boat is 0.384 m/s in the opposite direction of the thrown tackle box which is relative to the dock. Thus, the recoil speed of the fisherman and his boat is 0.384 m/s.

Key concepts

Momentum

Momentum is a fundamental concept in physics, representing the quantity of motion an object has. It is a vector quantity, meaning it has both magnitude and direction, and is defined as the product of an object's mass and its velocity. The formula for linear momentum (\textbf{p}) is expressed as \( \textbf{p} = m \times \textbf{v} \), where \( m \) is the mass and \( \textbf{v} \) is the velocity of the object.

In the given problem, understanding momentum helps to analyze the motion of the fisherman and the boat after the tackle box is thrown. Initially, the system is at rest; hence the total momentum is zero. The act of throwing the tackle box changes the distribution of momentum within the system, illustrating the principle of conservation of momentum: the total momentum before the event equals the total momentum after the event as long as no external forces are acting on the system.

In problems involving conservation of momentum, it is crucial to pay attention to the direction of the velocity vectors. Negative or positive signs in the computed velocities indicate the direction of motion relative to a chosen frame of reference, which in this case, is the dock.

Recoil Speed

Recoil speed refers to the velocity gained by an object moving in the opposite direction of another object being ejected or thrown away. This concept is commonly encountered in examples involving firearms, rockets, or any scenario where an object is expelled from a system, leading the remaining system to move in the converse direction due to the conservation of momentum.

When the fisherman throws the tackle box, the boat and the fisherman gain a velocity in the opposite direction. This velocity is the recoil speed. The magnitude of the recoil speed can be calculated using the conservation of momentum principle. The actual value of the recoil speed will depend on the mass of the objects involved as well as the speed of the expelled object. In the problem's context, the negative sign obtained in the final computation conveys that the fisherman and his boat move in the direction opposite to that of the thrown tackle box, aligning with the concept of recoil.

Physics Problem Solving

Identifying the Relevant Concept

The first step in solving any physics problem is to understand the concept involved. In the case of the fisherman and his boat, the concept is the conservation of momentum. Recognizing this allows you to set up the necessary equations properly.

Setting Up Equations

Once the concept is identified, the next step is translating the physical situation into a mathematical model — in this case, applying the conservation of momentum equation. The signs used in the equation reflect direction and should align with the chosen frame of reference.

Solving and Interpreting the Results

Substitute the given values into the equations and solve for the unknown. After obtaining the numerical answer, interpret it in the context of the question. A negative sign in velocity, for instance, may indicate direction rather than implying anything about the speed's magnitude.

Following these structured steps in physics problem-solving enhances understanding and accuracy.

Kinematics

Kinematics is the branch of mechanics that deals with the motion of objects without considering the forces causing the motion. It involves the description of motion in terms of displacement, velocity, and acceleration. In the context of the fisherman problem, understanding kinematics is vital as it deals with the velocities of moving objects. The primary kinematic equation relates to the velocity and the movement of the fisherman and his boat after throwing the tackle box.

Although the problem at hand does not delve into acceleration and displacement, kinematic principles allow us to understand how objects move and interact with each other after a particular kinematic event, such as the tackle box being thrown in this scenario. The problem leverages kinematic concepts to deduce the final velocities of the involved objects, which is a core aspect of kinematic analysis.