A 100 kg astronaut carries a launcher loaded with a 10 kg bowling ball; the launcher and the astronaut’s spacesuit have negligible mass. The astronaut discovers that firing the launcher results
in the ball moving away from her at a relative speed of 50 m/s.
The astronaut in the previous situation is now moving at 10 m/s (as measured in a certain frame
of reference). She wishes to fire the launcher so that her velocity turns through as large an angle
as possible (in this frame of reference). What is this maximum angle? (Hint: a diagram may be
useful.)
The answer is 27 degrees. I do not know how. I think she must fire the ball perpendicularly to her velocity, but I instead got 29 degrees as my answer. Help?
The maximum angle actually does not occur when she fires the ball perpendicularly to her velocity. It occurs when the angle between the velocity from the launcher and the resultant velocity are at 90 degrees to each other. The impulse of the astronaut is about 455 Ns. Therefore, the velocity given by the ball is 4.55 m/s. sin angle = 4.55/10 so the angle is 27 degrees.
50
To find the maximum angle at which the astronaut can fire the launcher, we need to consider the conservation of momentum and the law of cosines. Let's break down the problem step by step.
Step 1: Determine the initial momentum before firing the launcher.
The initial momentum of the astronaut and the launcher must be zero since there is no external force acting on them. Therefore, their total momentum is zero.
Initial momentum = Mass of the astronaut * Velocity of the astronaut + Mass of the ball * Velocity of the ball
Step 2: Determine the velocity of the astronaut and the launcher after firing the launcher.
Let's say after firing the launcher, the astronaut and the launcher move in opposite directions. So the astronaut's velocity will be in the opposite direction of the launcher ball's velocity.
Final momentum = Mass of the astronaut * (Velocity of the astronaut after firing) + Mass of the ball * (Velocity of the ball after firing)
Step 3: Apply the conservation of momentum
Since the total momentum before and after firing must be the same:
Initial momentum = Final momentum
Step 4: Solve for the velocity of the astronaut after firing
Setting up the equation and solving for the velocity of the astronaut after firing:
Mass of the astronaut * (Velocity of the astronaut) = Mass of the ball * (Velocity of the ball)
100 kg * (10 m/s) = 10 kg * (Velocity of the ball after firing)
Velocity of the ball after firing = 100 m/s
Step 5: Find the maximum angle
The maximum angle can be found by considering the relative velocity of the ball with respect to the astronaut. Given that the relative speed is 50 m/s, and the astronaut's velocity after firing is 100 m/s, we can define the angle using the law of cosines:
Relative speed^2 = (Velocity of the ball after firing)^2 + (Velocity of the astronaut after firing)^2 - 2 * (Velocity of the ball after firing) * (Velocity of the astronaut after firing) * cos(angle)
Solving for the angle:
angle = arccos[(Relative speed^2 - (Velocity of the ball after firing)^2 - (Velocity of the astronaut after firing)^2) / (-2 * (Velocity of the ball after firing) * (Velocity of the astronaut after firing))]
angle = arccos[(50^2 - 100^2 - 100^2) / (-2 * 100 * 100)]
angle = arccos[-18000 / -20000]
angle = arccos[0.9]
angle ≈ 27 degrees
Therefore, the maximum angle at which the astronaut can fire the launcher, so that her velocity turns through as large an angle as possible, is approximately 27 degrees.