Final answer:
The time taken for a ball to reach the wall in a room accelerating in free space is found by solving for the time in the quadratic equation resulting from the relative velocity of the ball with respect to the room. Two solutions are obtained, but only the first solution is physically meaningful for the scenario as it represents the time when the ball reaches the wall moving toward it.
Step-by-step explanation:
To calculate the time it takes for a ball to reach the wall in the given scenario, we first need to understand the relative motions of the room, the person, and the ball. The velocity of the ball with respect to the person throwing it is 5 m/s. However, since the room is accelerating, we need to account for the relative velocity of the ball with respect to the room.
The person is running at 2 m/s with respect to a fixed frame, and the room is accelerating in the direction of x at 2 m/s². When the ball is thrown, its initial velocity with respect to the room is the velocity with respect to the person plus the velocity of the person with respect to the room, which is 5 m/s + 2 m/s = 7 m/s. As time progresses, the acceleration of the room will affect the relative velocity, so we have vx,r = 7 - 2t.
We then use the equation for the position of the ball with respect to the room: xx,r = 7t - t². Setting up the equation for the distance to the wall being 10 m, we get the quadratic equation t² - 7t + 10 = 0. Solving for t, there are indeed two solutions, t = 2 and t = 5 seconds. The first solution corresponds to the time when the ball reaches the wall as it moves toward it, while the second solution actually describes the time when the ball would reach the same position if it moved in the opposite direction after bouncing back, under the assumption that it continued with similar motion. In reality, only the first solution is physically meaningful as the ball will hit the wall and not continue through in a straight line at the same relative velocity.