To determine the time it takes for the stone to reach the maximum height, we can use the equation for vertical motion:
v = u + at
where: v = final velocity (0 m/s at the highest point, as the stone momentarily stops before falling back down) u = initial velocity (25 m/s) a = acceleration (acceleration due to gravity, -9.8 m/s², negative because it acts in the opposite direction of the upward motion) t = time taken to reach the maximum height
At the maximum height, the final velocity is zero, so we can rewrite the equation as:
0 = 25 m/s - 9.8 m/s² * t_max
Solving for t_max:
9.8 m/s² * t_max = 25 m/s
t_max = 25 m/s / 9.8 m/s² t_max ≈ 2.55 seconds
Therefore, it takes approximately 2.55 seconds for the stone to reach the maximum height.
To find the maximum height (h), we can use the kinematic equation:
v^2 = u^2 + 2as
where: v = final velocity (0 m/s) u = initial velocity (25 m/s) a = acceleration (acceleration due to gravity, -9.8 m/s²) s = displacement (height)
Rearranging the equation, we have:
0 = (25 m/s)^2 + 2 * (-9.8 m/s²) * h
625 m²/s² = 19.6 m/s² * h
h = 625 m²/s² / (19.6 m/s²) h ≈ 31.88 meters
Therefore, the maximum height reached by the stone is approximately 31.88 meters.
To determine the time it takes for the stone to hit the ground, we can consider the time it takes to reach the same height as the launch point, but in the opposite direction. Since the initial velocity is 25 m/s upwards, the final velocity when the stone hits the ground will be -25 m/s (downwards).
Using the equation:
v = u + at
where: v = final velocity (-25 m/s) u = initial velocity (25 m/s) a = acceleration (acceleration due to gravity, -9.8 m/s²) t = time taken to hit the ground
-25 m/s = 25 m/s - 9.8 m/s² * t_ground
Solving for t_ground:
-9.8 m/s² * t_ground = -50 m/s
t_ground = -50 m/s / -9.8 m/s² t_ground ≈ 5.1 seconds
Therefore, it takes approximately 5.1 seconds for the stone to hit the ground.