An astronaut with a mass of 90 kg (including spacesuit and equipment) is drifting away from his spaceship at a speed o f.2 m/s with respect to the spaceship. The astronaut is equipped only with a 0.5 kg wrench to help him get back to the ship. With what speed and in what direction relative to the spaceship must he throw the wrench for his body to acquire a speed of .1 m/s and direct him back toward the spaceship?
Well, well, well, looks like this astronaut has found themselves in quite a pickle! Now, let's see if we can clown around with some physics to help them out.
To solve this problem, we need to apply the principle of conservation of momentum. If the astronaut wants to change their velocity, they have to throw something. In this case, it's the trusty 0.5 kg wrench.
Since the astronaut wants to head back towards the spaceship, they need to throw the wrench in the opposite direction. And since momentum is conserved, the momentum of the astronaut before throwing the wrench should be equal and opposite to the momentum of the astronaut (including the wrench) after throwing it.
Let's crunch some numbers (in the hope that we don't crunch any wrenches):
The initial momentum of the astronaut (including the wrench) is given by:
Momentum_before = (Mass_astronaut + Mass_wrench) * Velocity_astronaut
The final momentum of the astronaut (without the wrench) is given by:
Momentum_after = Mass_astronaut * Velocity_astronaut
And the final momentum of the wrench is:
Momentum_wrench = Mass_wrench * Velocity_wrench
Since momentum is conserved, we have:
Momentum_before = Momentum_after + Momentum_wrench
Plugging in the given values:
(90 kg + 0.5 kg) * 0.2 m/s = 90 kg * 0.1 m/s + 0.5 kg * Velocity_wrench
Simplifying:
(90.5 kg) * 0.2 m/s = 9 kg * m/s + 0.5 kg * Velocity_wrench
Now, since we want to find the velocity of the wrench, we can rearrange the equation:
0.5 kg * Velocity_wrench = (90.5 kg) * 0.2 m/s - 9 kg * m/s
Velocity_wrench = [(90.5 kg) * 0.2 m/s - 9 kg * m/s] / 0.5 kg
Calculating this:
Velocity_wrench = (18.1 kg*m/s - 9 kg*m/s) / 0.5 kg
Velocity_wrench = 18.1 m/s - 9 m/s
Velocity_wrench = 9.1 m/s
So, our intrepid astronaut needs to throw the wrench with a velocity of 9.1 m/s in the opposite direction of their current velocity, to bring themselves back to the spaceship at a speed of 0.1 m/s.
Just remember, my friend, don't throw any pies while performing this experiment. That could get very messy!
To solve this problem, we can use the principle of conservation of momentum. The total momentum before throwing the wrench must be equal to the total momentum after throwing it.
Let's assume that the astronaut's velocity after throwing the wrench is v and the wrench's velocity after being thrown is V.
Step 1: Calculate the initial momentum of the system.
The initial momentum consists of the astronaut's momentum and the wrench's momentum, both before being thrown. Since they are drifting away from the spaceship but the problem states that the velocity of the astronaut relative to the spaceship is 0.2 m/s, the astronaut's initial velocity with respect to the spaceship is -0.2 m/s. The wrench is initially at rest with respect to the spaceship, so its initial velocity with respect to the spaceship is 0 m/s.
Momentum of the astronaut = mass of the astronaut x velocity of the astronaut
Momentum of the wrench = mass of the wrench x velocity of the wrench
Momentum of the astronaut = 90 kg x (-0.2 m/s)
Momentum of the wrench = 0.5 kg x 0 m/s
Step 2: Calculate the final momentum of the system.
The final momentum consists of the astronaut's momentum and the wrench's momentum, both after being thrown. Since the astronaut wants to acquire a speed of 0.1 m/s and move back toward the spaceship, the astronaut's final velocity with respect to the spaceship is 0.1 m/s. The wrench's final velocity with respect to the spaceship is V.
Momentum of the astronaut = mass of the astronaut x velocity of the astronaut
Momentum of the wrench = mass of the wrench x velocity of the wrench
Momentum of the astronaut = 90 kg x 0.1 m/s
Momentum of the wrench = 0.5 kg x V
Step 3: Apply the principle of conservation of momentum.
According to the conservation of momentum, the initial momentum should be equal to the final momentum.
Momentum of the astronaut + Momentum of the wrench = Momentum of the astronaut + Momentum of the wrench
90 kg x (-0.2 m/s) + 0.5 kg x 0 m/s = 90 kg x 0.1 m/s + 0.5 kg x V
Simplify the equation:
-18 kg·m/s = 9 kg·m/s + 0.5 kg x V
Step 4: Solve for V.
Rearrange the equation and solve for V:
18 kg·m/s - 9 kg·m/s = 0.5 kg x V
9 kg·m/s = 0.5 kg x V
V = (9 kg·m/s) / 0.5 kg
V = 18 m/s
Therefore, the astronaut must throw the wrench with a velocity of 18 m/s in the opposite direction of the spaceship's motion (relative to the spaceship) for his body to acquire a speed of 0.1 m/s and direct him back toward the spaceship.
To solve this problem, we can apply the principle of conservation of momentum. According to this principle, the total momentum of an isolated system remains constant unless acted upon by external forces.
Let's break down the problem step by step:
Step 1: Calculate the initial momentum of the astronaut before throwing the wrench.
The initial momentum of the astronaut can be calculated using the formula: momentum = mass × velocity.
Given:
Mass of the astronaut (including spacesuit and equipment) = 90 kg
Speed of the astronaut relative to the spaceship = 0.2 m/s
Momentum of the astronaut (before throwing the wrench) = 90 kg × 0.2 m/s = 18 kg·m/s
Step 2: Calculate the final momentum required to return to the spaceship.
To propel himself back towards the spaceship, the astronaut needs to throw the wrench in the opposite direction. If he throws the wrench with a velocity, the total momentum of the system will be conserved.
We will assume that the astronaut and the wrench are the only objects involved in the system.
We'll denote the velocity of the astronaut after throwing the wrench as V_astronaut and the velocity of the wrench as V_wrench.
The final momentum of the astronaut and the wrench combined should be equal to the initial momentum (18 kg·m/s), which can be written as:
(90 + 0.5) kg × V_astronaut = 18 kg·m/s
Step 3: Calculate the speed and direction of the astronaut after throwing the wrench.
Rearranging the equation from Step 2:
V_astronaut = 18 kg·m/s / (90.5 kg)
V_astronaut ≈ 0.199 kg·m/s
The astronaut needs to have a velocity of approximately 0.199 m/s directed towards the spaceship.
Since the astronaut is initially drifting at a speed of 0.2 m/s away from the spaceship, he needs to throw the wrench with a slightly higher velocity to compensate for that drift and achieve a velocity of 0.199 m/s towards the spaceship.
The direction of the throw should be opposite to the current drift direction to counter it.