To tackle this problem, we need to analyze the motion of the ball shot from the elevator while considering the elevator's upward movement. We will break down the problem into manageable parts, calculating the time it takes for the ball to hit the floor of the elevator, the maximum height it reaches, its displacement with respect to the ground, and the maximum separation between the ball and the elevator floor. Let's dive into each part step by step.
1. Time Until the Ball Strikes the Elevator Floor
First, we need to determine how long it takes for the ball to hit the floor of the elevator. The ball is shot vertically upward with an initial velocity of 15 m/s relative to the elevator, which is moving upward at 10 m/s. Therefore, the ball's initial velocity relative to the ground is:
- Initial Velocity (u): 15 m/s (upward) + 10 m/s (elevator) = 25 m/s (upward)
The height from which the ball is shot is 2 m above the floor of the elevator. The equation of motion we will use is:
Here, h is the displacement of the ball, u is the initial velocity, g is the acceleration due to gravity (which we take as -10 m/s² since it acts downward), and t is the time in seconds. The ball will strike the floor of the elevator when it has fallen 2 m:
Rearranging gives:
Using the quadratic formula, t = [ -b ± √(b² - 4ac) ] / 2a, where a = 5, b = -25, and c = -2:
- t = [25 ± √(625 + 40)] / 10
- t = [25 ± √665] / 10
Calculating gives us two possible times, but we only consider the positive root:
2. Maximum Height Reached by the Ball from the Ground
Next, we calculate the maximum height the ball reaches above the ground. The maximum height occurs when the velocity of the ball becomes zero. Using the formula:
Setting v = 0 (at the maximum height), we have:
Solving for s (the height above the point of release):
Now, adding the initial height of 2 m from the floor of the elevator:
- Total Height = 31.25 m + 2 m = 33.25 m
Since the elevator is 50 m above the ground, the maximum height from the ground is:
- Maximum Height = 50 m + 33.25 m = 83.25 m
3. Displacement of the Ball with Respect to the Ground
The displacement of the ball with respect to the ground during its flight can be calculated by considering the total distance it travels upwards and downwards. The total vertical distance traveled is the maximum height reached minus the height of the elevator floor:
- Displacement = Maximum Height - Initial Height
- Displacement = 83.25 m - 50 m = 33.25 m
4. Maximum Separation Between the Elevator Floor and the Ball
The maximum separation occurs when the ball reaches its peak height. At this point, the elevator continues to rise. The height of the elevator at the time the ball reaches its maximum height can be calculated as:
- Height of Elevator = Initial Height + (Velocity of Elevator × Time)
- Height of Elevator = 50 m + (10 m/s × 2.13 s) = 50 m + 21.3 m = 71.3 m
The maximum separation is then the difference between the maximum height of the ball and the height of the elevator at that time:
- Maximum Separation = Maximum Height - Height of Elevator
- Maximum Separation = 83.25 m - 71.3 m = 11.95 m
However, if we consider the time it takes for the ball to reach its peak height (which is less than 2.13 seconds), we can refine this calculation. The ball reaches its peak height at about 2.5 seconds, so the maximum separation is approximately:
- Maximum Separation ≈ 9.5 m
In summary, we have calculated the time it takes for the ball to hit the elevator floor, the maximum height it reaches, its displacement with respect to the ground, and the maximum separation between the ball and the elevator floor during its flight. Each of these calculations relies on the principles of kinematics and the relative motion of the ball and the elevator.
