To find the speed at which the stone impacts the ground, we can use the concept of conservation of energy. Initially, the stone is at a height of 0 m and has potential energy. When it reaches the ground, its potential energy becomes zero, and all the initial potential energy is converted into kinetic energy.
The potential energy at the topmost point is given by the formula:
PE = mgh, where m is the mass of the stone, g is the acceleration due to gravity, and h is the height.
Given that g = 9.81 m/s^2, h = 12.7 m, and m is not provided, we can ignore the mass since it cancels out when calculating the speed at which the stone impacts the ground.
PE = mgh = 0.5mv^2, where v is the velocity at which the stone impacts the ground.
Solving for v, we get:
v = √(2gh)
Substituting the values, we have:
v = √(2 * 9.81 * 12.7) ≈ 15.8 m/s
Therefore, the speed at which the stone impacts the ground is approximately 15.8 m/s.
To find the time the stone is in the air, we can use the kinematic equation:
h = ut + 0.5gt^2.
Here, h is the height, u is the initial velocity, g is the acceleration due to gravity, and t is the time.
Since the stone is thrown upward and then falls downward, the total time in the air is twice the time it takes to reach the maximum height.
To find the time it takes to reach the maximum height, we use the equation:
u = v - gt,
where u is the initial velocity, v is the final velocity (0 m/s at the maximum height), g is the acceleration due to gravity, and t is the time.
0 = 6.05 - 9.81t.
Solving for t:
t = 6.05 / 9.81 ≈ 0.62 s.
Therefore, the time taken to reach the maximum height is approximately 0.62 seconds.
The total time in the air (falling downward) is twice that value:
Total time = 2 * 0.62 ≈ 1.24 seconds.
Therefore, the stone is in the air for approximately 1.24 seconds.