a 300 kg astronaut and equipment move with a velocity of 2.oom/s toward an orbiting spacecraft. How long must the astronaut fire a 100N rocket backpack to stop the motion relative to the spacecraft?
To solve this problem, we can use Newton's second law, which states that the force exerted on an object is equal to its mass multiplied by its acceleration:
F = m * a
In this case, the astronaut and equipment have a combined mass of 300 kg, and we need to find the time (let's call it t) for which the acceleration (a) should be applied to bring them to a stop.
The force required to stop the motion is given as 100 N. As acceleration is the rate of change of velocity, and we want to stop the motion, the final velocity (vf) will be 0 m/s. The initial velocity (vi) is given as 2.0 m/s.
We can use the following equation to find the acceleration:
a = (vf - vi) / t
Since we want to find the time for which the acceleration should be applied, we rearrange the equation:
t = (vf - vi) / a
Substituting the given values into the equation:
t = (0 - 2.0) / a
Now we need to calculate the acceleration. As the force (F) is given as 100 N and the mass (m) is 300 kg, we can use the formula:
F = m * a
100 = 300 * a
Solving for a:
a = 100 / 300
Substituting this value back into the equation for time:
t = (0 - 2.0) / (100 / 300)
Simplifying:
t = -2.0 / (100 / 300)
t = -2.0 * (300 / 100)
t = -6.0 seconds
Now, we have obtained a negative time value, which suggests that the equation was set up incorrectly. The negative sign indicates that the direction of acceleration is opposite to the direction of velocity, meaning the astronaut would need to decelerate to stop. However, since we're looking for a positive time, we can simply take the absolute value of the calculated time:
t = 6.0 seconds
Therefore, the astronaut must fire the rocket backpack for 6.0 seconds to stop the motion relative to the spacecraft.