A stone is thrown up with a velocity of 58 m/s from the ground. While coming down, it strikes a glass pan, held at half the height through which it has risen and loses half of its velocity in breaking the glass. Determine the time taken to reach the maximum height and the velocity with which it will strike the ground and total time taken to reach the ground
To solve this problem, we can break it down into two parts: the stone's motion while going up and its motion while coming down.
First, let's find the time taken to reach the maximum height.
We know that the initial velocity when the stone is thrown up is 58 m/s and the acceleration due to gravity is approximately 9.8 m/s².
We can use the kinematic equation:
v = u + at,
where:
v = final velocity,
u = initial velocity,
a = acceleration, and
t = time.
When the stone reaches the maximum height, its final velocity will be zero. So using the kinematic equation, we have:
0 = 58 - 9.8t.
Solving this equation for t will give us the time taken to reach the maximum height.
0 = 58 - 9.8t
9.8t = 58
t ≈ 5.92 seconds.
Therefore, it takes approximately 5.92 seconds to reach the maximum height.
Next, let's find the stone's velocity when it strikes the ground.
We know that when the stone strikes the glass pan, it loses half of its velocity. So the velocity after breaking the glass would be 58/2 = 29 m/s.
When the stone strikes the ground, its velocity would be the same as the velocity with which it hit after breaking the glass. So the velocity with which it will strike the ground is 29 m/s.
Finally, let's determine the total time taken to reach the ground.
The time to reach the maximum height is 5.92 seconds, and since the motion going up is symmetric to the motion coming down, the total time to reach the ground would be twice the time taken to reach the maximum height.
Therefore, the total time taken to reach the ground is approximately 2 * 5.92 = 11.84 seconds.