A 60-kg astronaut is in space, far from any objects that would exert a significant gravitational force on him. He would like to move toward his spaceship, but his jet pack is not functioning. He throws a 720-g socket wrench with a velocity of 5 m/s in a direction away from the ship. After 0.50 s, he throws an 800 g spanner in the same direction with a speed of 8 m/s. After another 9.90 s, he throws a mallet with a speed of 6 m/s in the same direction. The mallet has a mass of 1150 g. How fast is the astronaut moving after he throws the mallet?
To solve this problem, we need to understand the concept of momentum. Momentum is defined as the product of an object's mass and its velocity. The total momentum of a system is conserved if no external force acts on it.
We can calculate the total momentum of the astronaut before and after each object is thrown, and then use the principle of conservation of momentum to find the astronaut's final velocity.
Let's break it down step by step:
Step 1: Calculate the momentum before throwing the socket wrench.
The momentum of the astronaut before throwing the socket wrench is given by:
Momentum = mass × velocity = 60 kg × 0 m/s = 0 kg·m/s
Step 2: Calculate the momentum after throwing the socket wrench.
The momentum of the astronaut after throwing the socket wrench is given by:
Momentum = mass × velocity = (60 kg + 0.72 kg) × 5 m/s (velocity away from the spaceship)
= 60.72 kg × 5 m/s = 303.6 kg·m/s (velocity away from the spaceship)
Step 3: Calculate the momentum before throwing the spanner.
The momentum of the astronaut before throwing the spanner is equal to the momentum after throwing the socket wrench. This is because the astronaut is not acted upon by any external force, so the total momentum is conserved and does not change:
Momentum = 303.6 kg·m/s (velocity away from the spaceship)
Step 4: Calculate the momentum after throwing the spanner.
The momentum of the astronaut after throwing the spanner is given by:
Momentum = mass × velocity = (60 kg + 0.72 kg + 0.8 kg) × 8 m/s (velocity away from the spaceship)
= 61.52 kg × 8 m/s = 492.16 kg·m/s (velocity away from the spaceship)
Step 5: Calculate the momentum before throwing the mallet.
The momentum of the astronaut before throwing the mallet is equal to the momentum after throwing the spanner, as the total momentum is conserved:
Momentum = 492.16 kg·m/s (velocity away from the spaceship)
Step 6: Calculate the momentum after throwing the mallet.
The momentum of the astronaut after throwing the mallet is given by:
Momentum = mass × velocity = (60 kg + 0.72 kg + 0.8 kg + 1.15 kg) × 6 m/s (velocity away from the spaceship)
= 62.67 kg × 6 m/s = 376.02 kg·m/s (velocity away from the spaceship)
Step 7: Calculate the final velocity of the astronaut.
To find the final velocity of the astronaut, we need to use the principle of conservation of momentum. Since the total momentum before throwing the mallet is the same as the total momentum after throwing the mallet, we can set these two values equal to each other:
492.16 kg·m/s = 376.02 kg·m/s + 60 kg × vf (where vf is the final velocity of the astronaut)
Rearranging the equation to isolate vf, we get:
vf = (492.16 kg·m/s - 376.02 kg·m/s) / 60 kg
vf = 2.1033 m/s (approximately)
Therefore, the astronaut's final velocity after throwing the mallet is approximately 2.1033 m/s away from the spaceship.