A 200.0 kg astronaut and equipment move with a velocity of 2.00 m/s toward an orbiting spacecraft. How long will the astronaut need to fire a 100.0 N rocket backpack to stop the motion relative to the spacecraft?
100*timefiring= mass*changevelocity
To find the time the astronaut needs to fire the rocket backpack, we first need to find the acceleration caused by the 100 N force. We can use the formula:
Force = mass * acceleration
100 N = 200 kg * acceleration
Now we can solve for acceleration:
acceleration = 100 N / 200 kg
acceleration = 0.5 m/s^2
Next, we can use the formula for acceleration to find the time it takes to come to a stop:
change in velocity = acceleration * time
-2 m/s = 0.5 m/s^2 * time
Now we can solve for time:
time = -2 m/s / 0.5 m/s^2
time = -4 s
However, we cannot have a negative time, so the astronaut should fire the rocket backpack in the opposite direction:
time = 4 s
So the astronaut needs to fire the rocket backpack for 4 seconds to stop the motion relative to the spacecraft.
To calculate the time the astronaut needs to fire the rocket backpack to stop the motion relative to the spacecraft, we can use the equation:
100 * time_firing = mass * change_in_velocity
Given:
- mass of the astronaut and equipment (m) = 200.0 kg
- velocity of the astronaut and equipment (v) = 2.00 m/s
- force applied by the rocket backpack (F) = 100.0 N
First, we need to calculate the change in velocity (Δv):
Δv = -v
= -2.00 m/s
Substituting the values into the equation:
100 * time_firing = 200.0 kg * (-2.00 m/s)
Simplifying:
100 * time_firing = -400.0 kg * m/s
Now, we can solve for time_firing by dividing both sides of the equation by 100:
time_firing = (-400.0 kg * m/s) / 100
time_firing = -4.00 s
Since time cannot be negative, we discard the negative sign and the time taken to fire the rocket backpack to stop the motion relative to the spacecraft is 4.00 seconds.
To determine the time required for the astronaut to stop the motion relative to the spacecraft, we can use the principle of conservation of momentum. This principle states that the total momentum before an event is equal to the total momentum after the event, assuming no external forces are acting on the system.
In this case, the initial momentum of the astronaut and equipment moving toward the spacecraft is given by the product of their mass and velocity:
Initial momentum = (mass of astronaut and equipment) x (initial velocity)
Final momentum, after the astronaut fires the rocket backpack and stops the motion, is zero because the astronaut will be at rest relative to the spacecraft:
Final momentum = 0
We can set up an equation using the principle of conservation of momentum:
Initial momentum - Final momentum = change in momentum
(mass of astronaut and equipment) x (initial velocity) - 0 = (mass of astronaut and equipment) x (final velocity)
Substituting the given values, we have:
(200.0 kg) x (2.00 m/s) = (200.0 kg) x (final velocity)
Simplifying the equation, we find:
400.0 kg⋅m/s = 200.0 kg⋅(final velocity)
Dividing both sides of the equation by 200.0 kg, we get:
(final velocity) = 400.0 kg⋅m/s / 200.0 kg
(final velocity) = 2.00 m/s
Now we know the final velocity that the astronaut needs to achieve in order to stop the motion.
To calculate the time required for the astronaut to fire the rocket backpack, we can use Newton's second law of motion, which states that the force acting on an object is equal to the product of its mass and acceleration:
Force = mass x acceleration
In this case, the force exerted by the rocket backpack is given as 100.0 N, and the mass of the astronaut is 200.0 kg.
Substituting these values into the equation, we have:
100.0 N = 200.0 kg x (final velocity - initial velocity) / time
Since the initial velocity is 2.00 m/s and the final velocity is 0 m/s (rest), we can simplify the equation:
100.0 N = 200.0 kg x (-2.00 m/s) / time
To solve for the time, we can rearrange the equation:
time = 200.0 kg x (-2.00 m/s) / 100.0 N
Simplifying the equation, we get:
time = -4.00 kg⋅m/s / 100.0 N
time = -0.04 s
The time needed for the astronaut to fire the 100.0 N rocket backpack to stop the motion relative to the spacecraft is approximately 0.04 seconds.