Answer:
We can use the kinematic equations of motion to solve this problem. Let's assume that the initial velocity of the stone is v0, and that it is thrown at an angle θ above the horizontal. We can then break down the initial velocity into its x and y components:
v0x = v0 cosθ
v0y = v0 sinθ
We know that one second after the stone is thrown, its velocity in the y direction is 8 m/s, and that it is moving downwards. Using the equation of motion for free fall, we can find the acceleration due to gravity:
v = v0 + at
8 m/s = v0 sinθ - 9.8 m/s^2
Solving for v0 sinθ, we get:
v0 sinθ = 8 m/s + 9.8 m/s^2
v0 sinθ = 17.8 m/s
We can also use the time of flight equation to find the time it takes for the stone to reach its maximum height:
t = 2v0 sinθ / g
Solving for t, we get:
t = 2(17.8 m/s) sinθ / 9.8 m/s^2
t = 3.64 s
The maximum height can then be found using the equation for vertical displacement:
y = v0y t - 1/2 g t^2
At the maximum height, the velocity of the stone is zero, so we can set v0y t - 1/2 g t^2 = 0 and solve for y:
y = v0y^2 / (2g)
Substituting the known values, we get:
y = (v0 sinθ)^2 / (2g)
y = (17.8 m/s)^2 / (2 * 9.8 m/s^2)
y = 16.1 m
Therefore, the initial velocity of the stone is approximately v0 = 19.4 m/s, and its maximum height is approximately 16.1 m.