Question

A fisherman is sitting in his 28.51 -kg fishing boat along with his 14.91 -kg tackle box. The boat and its cargo are at rest near a dock. He throws the tackle box toward the dock, which it approaches with a speed of \(3.303 \mathrm{~m} / \mathrm{s}\). The fisherman and his boat recoil with a speed of \(0.4547 \mathrm{~m} / \mathrm{s}\). What is the mass of the fisherman?

Short answer

Answer: The mass of the fisherman is approximately 154.82 kg.

Step-by-step solution

Determine Initial Momentum

Before the fisherman throws the tackle box, both the boat and tackle box are at rest. So, their initial velocities are 0. Hence, the initial total momentum is also 0.

Determine Final Momentum

After the tackle box is thrown, the boat (with the fisherman in it) and the tackle box have different final velocities. We are given the final velocities of both the boat and the tackle box, and their respective masses. We can calculate their momenta by multiplying their masses and velocities, and then find the total final momentum by adding these momenta. Total final momentum = (mass of tackle box × velocity of tackle box) + (mass of boat × velocity of boat) + (mass of fisherman × velocity of fisherman)

Apply Conservation of Momentum

According to the law of conservation of momentum, the sum of the initial momenta equals the sum of the final momenta. So our equation becomes: 0 = (14.91 kg × 3.303 m/s) + (28.51 kg × -0.4547 m/s) + (mass of fisherman × -0.4547 m/s)

Solve for the Mass of the Fisherman

Now, we can solve the equation for the mass of the fisherman: mass of fisherman × -0.4547 m/s = -(14.91 kg × 3.303 m/s) - (28.51 kg × -0.4547 m/s) Divide both sides by -0.4547 m/s to isolate the mass of the fisherman: mass of fisherman = \(\frac{-(14.91 kg × 3.303 m/s) - (28.51 kg × -0.4547 m/s)}{-0.4547 m/s}\) mass of fisherman ≈ 154.82 kg So, the mass of the fisherman is approximately 154.82 kg.

Key concepts

Linear Momentum

Understanding the concept of linear momentum is crucial in physics, especially when examining the movement of objects. It's a measure of the amount of motion an object carries and is a product of an object's mass and velocity. That is to say, the momentum of an object (\( p \) ) can be calculated using the formula \[ p = m \times v \] where \( m \) is the mass of the object and \( v \) is its velocity.

Imagine you're pushing a shopping cart; the harder you push (increasing the cart's velocity), or the more items you add to the cart (increasing the mass), the greater the cart’s momentum. This means the cart will be harder to stop, showing that both mass and velocity are fundamental in determining the momentum of an object.

For example, in the exercise with the fisherman throwing his tackle box, each object—the tackle box, the boat, and the fisherman himself—has its own linear momentum after the box is thrown. This is why the fisherman and his boat start moving; the throwing action imparts momentum to the tackle box, and to conserve momentum, the boat moves in the opposite direction.

Momentum Conservation Law

Momentum Conservation Law forms a cornerstone in the study of physics, particularly in the domain of mechanics. It states that the total linear momentum of a closed system remains constant if no external forces act upon it. In other words, momentum cannot be created or destroyed but can only be transferred from one object to another.

Applying this principle to real life, consider a game of billiards. When the cue ball strikes another ball, the cue ball slows down while the other ball speeds up. The momentum lost by the cue ball is transferred to the second ball, but the total momentum of both balls before and after the collision remains the same.

In the context of the fisherman and his tackle box, before the box is thrown, their combined momentum is zero because they're not moving. Once the fisherman throws the tackle box, the momentum of the box and the boat (with the fisherman in it) must sum up to zero to satisfy the conservation of momentum, allowing us to calculate his mass.

Mass and Velocity

Mass and velocity are fundamental concepts in the equation of momentum. Mass is a measure of the amount of matter in an object and remains constant regardless of its location and the speed at which it's moving. Velocity, on the other hand, is the speed of an object in a given direction and can change depending on external forces and actions.

Together, mass and velocity determine the momentum of an object. In everyday examples, a moving car (with significant mass and velocity) has more momentum compared to a feather being gently pushed by the breeze. Moreover, if an object's velocity changes due to an acceleration, its momentum changes proportionately.

The fisherman's problem illustrates these concepts nicely as we see that the mass of the tackle box and the boat remains constant, while their velocities change when the tackle box is thrown. The resulting change in velocities is essential for solving the problem as it leads to finding the unknown mass of the fisherman by using the principle of momentum conservation.