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.

Step by step solution

01

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.

02

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

03

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

04

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#

05

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#

06

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

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