A boy throws a stone vertically up to a man standing at a heigt of 2 m above the boy. Of the stone is thrown up with a velocity of 7.0m s what is the velocity of the stone at the instant when it is caught by the man.
A clear explaination is greatly appreciated
Kinetic energy of M*g*H is lost, with H = 2.0 m.
The initial Velocity is
V1 = 7.0 m/s
The final kinetic energy is
(1/2) M V2^2 = (1/2) M V1^2 - M g H
Cancel out the M's and solve for the final velocity V2.
V2^2 - V1^2 = -2 g H
Does that give me a negative velocity?
To determine the velocity of the stone when it is caught by the man, we need to apply the principles of motion under gravity.
We can break down the problem into two parts: the stone moving upwards and the stone moving downwards.
1. Stone moving upwards:
Initially, the stone has an initial velocity of 7.0 m/s in the upward direction. As it moves against the force of gravity, the velocity gradually decreases. At the highest point, also known as the peak or the maximum height, the velocity becomes zero momentarily before reversing direction and coming back down.
To determine the time it takes for the stone to reach the peak, we can use the equation:
Vf = Vi + at
Where:
- Vf is the final velocity (which is zero at the peak)
- Vi is the initial velocity (7.0 m/s upwards)
- a is the acceleration experienced due to gravity (-9.8 m/s^2, considering downward motion)
Rearranging the equation, we can solve for the time taken to reach the peak:
0 = 7.0 - 9.8t
Solving for t:
9.8t = 7.0
t ≈ 0.714 seconds
So, it takes approximately 0.714 seconds for the stone to reach the peak.
2. Stone moving downwards:
Once the stone reaches the peak, it starts accelerating downwards due to the force of gravity. At the instant when the man catches the stone, the stone and the man are at the same height.
To calculate the velocity of the stone when it is caught, we can use the equation:
Vf = Vi + at
Where:
- Vf is the final velocity (what we want to determine)
- Vi is the initial velocity (which was zero at the peak)
- a is the acceleration experienced due to gravity (-9.8 m/s^2, still considering downward motion)
Rearranging the equation, we can solve for the final velocity:
Vf = -9.8t
Substituting the value of t (0.714 seconds):
Vf = -9.8 × 0.714
Vf ≈ -7.0 m/s
Since the velocity is negative, it indicates that the stone is moving downwards when it is caught by the man.
Therefore, the velocity of the stone at the instant it is caught by the man is approximately -7.0 m/s (downward direction).