7. An airplane has a gun that fires bullets straight ahead at the speed of 600 miles per hour when tested on the ground while the plane is stationary. The plane takes off and flies east at 600 miles per hour. Explain whether each of the following is possible for the pilot to measure:
a. When fired directly ahead the bullets move eastward at a speed of 1200 miles per hour
b. When fired in the opposite direction (by turning the gun westward, but not the plane), the bullets drop vertically downward
c. If fired vertically downward (by turning the gun downward, but not the plane), the bullets move eastward at 600 miles per hour while they fall.
Unless the pilot can track a speeding bullet, I don't see how he can take any of those measurements.
I guess he could turn the gun backwards and fire it when passing over an observer. If the observer sees the bullets come straight down, the speeds are ok.
a. It is not possible for the pilot to measure bullets moving eastward at a speed of 1200 miles per hour when fired directly ahead while the plane is flying eastward at 600 miles per hour. The reason for this is due to the principle of the conservation of momentum. In this case, when the gun is fired, the bullets will inherit the forward velocity of the plane, which means they will already be moving eastward at 600 miles per hour with respect to the ground. Therefore, the bullets cannot exceed this speed as there is no additional force acting on them to make them move faster.
To verify this, one can use the principle of velocity addition. When two velocities in the same direction are added, the resulting velocity is the sum of the individual velocities. In this case, if the bullets were to move eastward at a speed of 1200 miles per hour, it would mean that the bullets are traveling at 600 miles per hour relative to the plane, and an additional 600 miles per hour relative to the ground. However, this violates the principle of velocity addition, so it is not possible.
b. It is not possible for the bullets to drop vertically downward when fired in the opposite direction (westward) without the plane turning. The reason is that in the absence of external forces, objects in motion will continue to move with a constant velocity. In this scenario, if the gun is fired westward while the plane is flying eastward, the bullets will still inherit the forward velocity of the plane. Therefore, the bullets will continue to move eastward with the same speed as the plane, instead of dropping vertically downward.
c. If the gun is turned downward (but not the plane) and fired vertically downward, the bullets will free-fall due to the force of gravity acting on them. During this free-fall, the bullets will continue to move eastward at an initial speed of 600 miles per hour, the same as the plane's velocity. However, as the bullets fall, their vertical speed will increase due to the pull of gravity. This means that the horizontal velocity component (eastward) stays constant at 600 miles per hour, while the vertical velocity component (downward) increases steadily due to gravity's pull.