Question

You find yourself in the (realistic?) situation of being stuck on a 300 -kg raft (including yourself) in the middle of a pond with nothing but a pile of 7 -kg bowling balls and 55 -g tennis balls. Using your knowledge of rocket propulsion, you decide to start throwing balls from the raft to move toward shore. Which of the following will allow you to reach the shore faster? a) throwing the tennis balls at \(35 \mathrm{~m} / \mathrm{s}\) at a rate of 1 tennis ball per second b) throwing the bowling balls at \(0.5 \mathrm{~m} / \mathrm{s}\) at a rate of 1 bowling ball every \(3 \mathrm{~s}\) c) throwing a tennis ball and a bowling ball simultaneously, with the tennis ball moving at \(15 \mathrm{~m} / \mathrm{s}\) and the bowling ball moving at \(0.3 \mathrm{~m} / \mathrm{s}\), at a rate of 1 tennis ball and 1 bowling ball every \(4 \mathrm{~s}\) d) not enough information to decide

Short answer

Answer: a) Throwing a 55g tennis ball at 35 m/s every second.

Step-by-step solution

Option a)

Calculate the momentum transferred by throwing a 55g tennis ball at 35 m/s every second: 1. Convert the mass of the tennis ball to kg: \(55 \mathrm{~g} = 0.055 \mathrm{~kg}\) 2. Calculate the momentum: \(p = mv = 0.055\mathrm{~kg} \times 35\mathrm{~m/s} = 1.925 \mathrm{~kg \cdot m/s}\) 3. Since the rate of transfer is 1 ball per second, the net momentum transfer per second is equal to the momentum of one ball: \(1.925 \mathrm{~kg \cdot m/s}\).

Option b)

Calculate the momentum transferred by throwing a 7-kg bowling ball at 0.5 m/s every 3 seconds: 1. Calculate the momentum: \(p = mv = 7\mathrm{~kg} \times 0.5\mathrm{~m/s} = 3.5 \mathrm{~kg \cdot m/s}\) 2. Calculate the transfer rate per second: \(\frac{3.5 \mathrm{~kg \cdot m/s}}{3\mathrm{~s}} = 1.1667 \mathrm{~kg \cdot m/s}\).

Option c)

Calculate the momentum transferred by throwing a tennis ball and a bowling ball simultaneously at the given speeds, every 4 seconds: 1. Calculate the momentum of the tennis ball: \(p_t = mv = 0.055\mathrm{~kg} \times 15\mathrm{~m/s} = 0.825 \mathrm{~kg \cdot m/s}\) 2. Calculate the momentum of the bowling ball: \(p_b = mv = 7\mathrm{~kg} \times 0.3\mathrm{~m/s} = 2.1 \mathrm{~kg \cdot m/s}\) 3. Calculate the total momentum: \(p_t + p_b = 0.825 \mathrm{~kg \cdot m/s} + 2.1 \mathrm{~kg \cdot m/s} = 2.925 \mathrm{~kg \cdot m/s}\) 4. Calculate the transfer rate per second: \(\frac{2.925 \mathrm{~kg \cdot m/s}}{4\mathrm{~s}} = 0.73125 \mathrm{~kg \cdot m/s}\). Now let's compare the momentum transfer rates for each option: Option a) \(1.925 \mathrm{~kg \cdot m/s}\) Option b) \(1.1667 \mathrm{~kg \cdot m/s}\) Option c) \(0.73125 \mathrm{~kg \cdot m/s}\) Option a) provides the highest momentum transfer rate, which means that throwing the tennis balls at 35 m/s with a rate of 1 ball per second will allow the student to reach the shore faster. So the answer is a).

Key concepts

Momentum Transfer in Rocket Propulsion

When you're stranded on a raft with nothing but balls to throw for propulsion, as in our exercise, an understanding of momentum transfer becomes your lifeline. Momentum transfer refers to the amount of momentum that moves from one object to another. In the case of the raft, by throwing balls, you're transferring momentum to the water through the reaction force.

Since every action has an equal and opposite reaction (Newton's third law), throwing an object from the raft will cause the raft itself to move in the opposite direction. The mass of the object and its velocity determine how much momentum is transferred. For the stranded student with a need for speed, the goal is to maximize this transfer.

To visualize this, imagine each thrown ball pushes the raft just a bit towards safety. The faster and heavier the ball, the harder the push. The transferred momentum (\( p = mv \text{, where } m \text{ is mass and } v \text{ is velocity}\) ) is what propels the raft. However, it's not just about a single throw; it's the continuous transfer over time that builds up speed and distance.

Conservation of Momentum and Its Effects

Conservation of momentum is a fundamental principle in physics that states that within a closed system, the total momentum remains constant if no external forces are acting on it. The raft-ball system in our exercise is a great example of this principle at work.

When a ball is thrown, the momentum of the ball and the raft-to-shore system must stay the same before and after the throw. Since the system initially has zero momentum (because it’s stationary), the momentum gained by the raft will be equal in magnitude and opposite in direction to the momentum the thrown ball has.

This concept helps us understand why option a) in the exercise allows the raft to reach the shore faster. It offers the greatest rate of momentum change per second, crucial for maximizing propulsion efficiency within the constraints of a closed system.

Physics Problem Solving Techniques

Physics problem solving often involves breaking down a scenario into measurable quantities to apply relevant principles and formulas. In our raft conundrum, we have employed a systematic approach for each possibility outlined in the question.

Firstly, we identified the quantity we needed to compare: momentum transfer rate. Next, we set up the problem by applying the formula for momentum (\( p = mv \text{, with } p \text{ as momentum, } m \text{ as mass and } v \text{ as velocity}\) ) for each thrown object. Then, by considering the given rates at which the objects are thrown, we were able to calculate momentum transfer rate per second.

Always focus on understanding the concepts involved: here, it's rocket propulsion via conservation and transfer of momentum. Evaluate how the numbers interact under these concepts. Step by step, this methodical process will unfold the solution and deepen your comprehension, ensuring those concepts stick with you long after you've left the raft.