Question

An astronaut of mass \(M\) is floating in space at a constant distance \(D\) from his spaceship when his safety line breaks. He is carrying a toolbox of mass \(M / 2\) that contains a big sledgehammer of mass \(M / 4\), for a total mass of \(3 M / 4\). He can throw the items with a speed \(v\) relative to his final speed after each item is thrown. He wants to return to the spaceship as soon as possible. a) To attain the maximum final speed, should the astronaut throw the two items together, or should he throw them one at a time? Explain. b) To attain the maximum speed, is it best to throw the hammer first or the toolbox first, or does the order make no difference? Explain. c) Find the maximum speed at which the astronaut can start moving toward the spaceship.

Short answer

- It is better for the astronaut to throw the items one at a time, as it results in a higher final speed (4v/3) compared to throwing them together (3v/4). b) Does the order in which the astronaut throws the items make a difference? - No, the order in which the astronaut throws the items does not make a difference. The final speed is the same (4v/3) whether the sledgehammer is thrown first or the toolbox is thrown first. c) What is the maximum speed at which the astronaut can start moving towards the spaceship? - The maximum speed at which the astronaut can start moving towards the spaceship is 4v/3 when the items are thrown one at a time with a relative speed of v.

Step-by-step solution

Determine initial momentum

Initially, the astronaut and his tools are at rest, so their initial total momentum is zero. Therefore, after throwing the items, the final total momentum must also be zero.

Calculate final speed when items are thrown together

We consider the sledgehammer of mass M/4 and the toolbox of mass M/2 as a single object with mass 3M/4. When they are thrown together with speed v, the final momentum of the astronaut is equal and opposite to the thrown mass's momentum. Thus, the astronaut's final speed is: Vm_a = (3M/4) * v / M = 3v/4

Calculate final speed when items are thrown one at a time

First, assume that the astronaut throws the sledgehammer of mass M/4 with speed v. After throwing the sledgehammer, the momentum of the astronaut is equal and opposite to the sledgehammer's momentum. Thus, the astronaut's final speed after throwing sledgehammer is: Vm_sh = (M/4) * v / (3M/4) = v/3 Next, the astronaut throws the toolbox of mass M/2 with speed v relative to his current speed Vm_sh. The astronaut's final speed after throwing the toolbox is: Vm_tb = (M/2) * v / (M/2) = v Adding the speeds from throwing the sledgehammer and the toolbox, we get the total final speed: Vm_b = Vm_sh + Vm_tb = v/3 + v = 4v/3

Compare final speeds when items are thrown together or one at a time

Compare the final speeds: Vm_a = 3v/4 Vm_b = 4v/3 Since 4v/3 > 3v/4, it is better to throw the items one at a time to attain the maximum final speed. #b) Determining the optimal order to throw the objects#

Determine if the order makes a difference

We have already determined that throwing the items one at a time results in the maximum final speed. Therefore, simply check whether throwing the sledgehammer first or the toolbox first yields the same result: When the astronaut throws the sledgehammer first, his final speed is 4v/3 (as calculated in Step 3). Now, imagine the astronaut throws the toolbox first: Vm_tb_first = (M/2) * v / (M/2) = v Vm_sh_first = (M/4) * v / (3M/4) = v/3 Adding the speeds, we obtain the same result: Vm_order = Vm_tb_first + Vm_sh_first = v + v/3 = 4v/3 Since the final speeds are the same in both cases, the order in which the items are thrown makes no difference. #c) Calculating the maximum speed of the astronaut#

Calculate the maximum speed using previous results

Using the results from Step 4, the maximum speed at which the astronaut can start moving towards the spaceship is achieved by throwing the items one at a time with a relative speed of v: Vm_max = 4v/3

Key concepts

Astronaut dynamics

In the vast expanse of space, understanding the dynamics of an astronaut is crucial for exploring the uncharted territories effectively. Let’s imagine an astronaut floating in space without any external forces acting, which is a common scenario.

• At the heart of astronaut dynamics is the principle of conservation of momentum, stating that the total momentum of the system must remain constant if no external forces are acting upon it.
• In the given problem, the astronaut starts at rest, meaning his initial momentum is zero. Any movement, like throwing objects, must ensure the total momentum remains zero.
• This principle is what the astronaut leverages to move back to the spaceship after his safety line breaks.

Understanding these dynamics helps astronauts maneuver in space, especially when untethered from the spacecraft.

Impulse and momentum

Impulse and momentum are intertwined concepts that play a significant role in understanding movements in physics, particularly in a gravity-free environment. Momentum is the product of an object's mass and velocity, given by the formula:\[ p = mv \]Where:

• \( p \) is the momentum,
• \( m \) is the mass,
• \( v \) is the velocity.

Impulse, on the other hand, is the change in momentum caused by an external force over a specific time period. In the scenario of our astronaut:

• The impulse exerted when he throws the objects results in an equal and opposite change in velocity, allowing him to propel towards safety.
• The astronaut's goal is to maximize this change in momentum to return to his spacecraft as swiftly as possible. By strategically throwing the objects one by one, he increases his velocity more than if he were to throw them simultaneously.

This approach allows for maximizing the effect of the impulse, thereby optimizing his return journey.

Problem solving in physics

Problem solving in physics often involves applying core principles effectively to derive solutions. In the presented problem:

• The astronaut employs conservation of momentum and strategic thinking to maximize his velocity towards the spaceship.
• He must decide whether to throw the sledgehammer and toolbox together or separately, and in which order, to achieve the greatest speed.

Important steps in problem solution include:

• Understanding the constraints and options available.
• Using mathematical equations to calculate potential outcomes, as shown in the detailed step-by-step calculations.
• Comparatively analyzing the outcomes to derive the most effective solution.

By considering single and sequential object throwing scenarios, he calculated the ultimate velocity potential. The strategy was validated through comparative mathematics, confirming that staggering throws optimizes his movement outcome. Hence, such exercises bolster critical thinking and problem-solving skills crucial across diverse physics applications.