2. A ball is thrown vertically upward from the 15m level in an elevator shaft with an initial velocity of 20 m/s. At the same instant an open-platform elevator passes the 6m level, moving upward with a constant velocity of 2 m/s. Determine
a. When and where the ball will hit the elevator,
b. The relative velocity of the ball with respect to the elevator when the ball hits the elevator.
To determine when and where the ball will hit the elevator, we need to analyze the vertical motion of the ball and the elevator separately and then find the intersection point.
a. When and where the ball will hit the elevator:
1. First, let's find the time it takes for the ball to reach its highest point. To do this, we can use the kinematic equation for vertical motion:
v = u + at, where v is the final velocity, u is the initial velocity, a is the acceleration, and t is the time.
Since the ball is moving against gravity, the acceleration is -9.8 m/s^2 (negative because it acts downward), and the initial velocity is 20 m/s (upward). At the highest point, the final velocity is 0 m/s.
Therefore, we can rearrange the equation to solve for the time:
t = (v - u) / a
t = (0 - 20) / (-9.8)
t ≈ 2.04 seconds
2. Since the ball takes the same time to reach the highest point as it does to fall back down and hit the elevator, we can double the time to get the total time of flight:
t_total = 2 * t
t_total ≈ 2 * 2.04
t_total ≈ 4.08 seconds
3. Now, let's find the height at which the ball will hit the elevator. We use the equation for vertical motion, considering the initial velocity, time, and acceleration.
s = ut + (1/2)at^2
u = 20 m/s (upward)
t = t_total ≈ 4.08 seconds
a = -9.8 m/s^2 (downward)
Plugging in the values:
s = (20)(4.08) + (1/2)(-9.8)(4.08)^2
s ≈ 81.6 - 83.858
s ≈ -2.258 meters (negative because it is below the starting point)
The ball will hit the elevator at approximately 2.258 meters below the starting point (15m level) in the elevator shaft.
b. The relative velocity of the ball with respect to the elevator when the ball hits the elevator:
Since the elevator is moving upward, we need to consider its velocity when calculating the relative velocity of the ball.
1. The velocity of the ball just before hitting the elevator can be calculated using the equation:
v = u + at
u = 20 m/s (upward)
a = -9.8 m/s^2 (downward)
t = t_total ≈ 4.08 seconds
Plugging in the values:
v = 20 - 9.8(4.08)
v ≈ 20 - 39.984
v ≈ -19.984 m/s (negative because it is downward)
2. To find the relative velocity, we subtract the velocity of the elevator from the velocity of the ball:
relative velocity = v_ball - v_elevator
v_elevator = 2 m/s (upward)
Plugging in the values:
relative velocity ≈ -19.984 - 2
relative velocity ≈ -21.984 m/s (negative because it is downward)
The relative velocity of the ball with respect to the elevator when the ball hits the elevator is approximately -21.984 m/s (downward).