A stone is thrown straight up and reaches a maximum height of 26 m. With what speed was the stone thrown?
V^2 = Vo^2 + 2g*h
V = 0 @ max h.
g = -9.8 m/s^2.
h = 26 m.
Vo = ? (m/s).
V(initial)=*2(g)(h)*
**<--- substitute for a square root symbol
22.5743217
To determine the speed at which the stone was thrown, we can use the equations of motion for objects in free fall. Since the stone reaches a maximum height and then falls back down, we can assume it experiences constant acceleration due to gravity.
We're given that the stone reaches a maximum height of 26 m, which means it traveled a total vertical distance of 26 m upwards and then 26 m downwards. The total distance traveled will be the sum of the upward and downward distances.
The upward distance is the same as the downward distance, so let's calculate the distance traveled in one direction. Using the equation for distance traveled in free fall:
s = ut + (1/2)at^2
where:
s = distance traveled
u = initial velocity (throw velocity)
t = time taken
a = acceleration (due to gravity, which is approximately 9.8 m/s^2)
Initially, the stone is thrown upwards and then falls back down, so the time taken will be twice the time taken to reach the maximum height. This is because the stone goes up and comes back down without any air resistance.
Let's denote the time taken to reach the maximum height as 't_up'. We have:
26 = u * t_up - (1/2) * 9.8 * t_up^2 (equation 1)
The stone then falls back down from the maximum height, so the total time taken will be:
t = 2 * t_up
Now, let's solve equation 1 for 't_up' by rearranging it:
t_up = (v_f - u) / a
where:
v_f is the final velocity, which is 0 m/s at the topmost point.
0 = u - 9.8 * t_up
Rearranging the equation, we can solve for 't_up':
t_up = u / 9.8 (equation 2)
Substituting equation 2 into the equation t = 2 * t_up:
t = 2 * (u / 9.8)
Now, let's plug in the known values:
26 = u * 2 * (u / 9.8)
Simplifying:
26 = 2u^2 / 9.8
Solving for u:
u^2 = (26 * 9.8) / 2
u^2 = 127.4
Taking the square root of both sides:
u = √127.4
u ≈ 11.3 m/s
Therefore, the stone was thrown with a speed of approximately 11.3 m/s.