When a ball is thrown vertically upwards, its initial velocity (u) can be determined using the time it takes to reach the maximum height (t) and the acceleration due to gravity (g).
In this case, the ball takes 3 seconds to reach the maximum height. At the maximum height, the vertical velocity becomes zero before it starts falling back down. Thus, the time taken to reach the maximum height is equal to the time taken to fall back to the initial position.
Using the equation for displacement in vertical motion:
S = ut + (1/2)gt^2
where: S = displacement (change in height) u = initial velocity t = time taken
At the maximum height, the displacement is zero. Therefore, the equation becomes:
0 = ut + (1/2)gt^2
We know that the acceleration due to gravity is approximately -9.8 m/s² (negative because it acts downwards). Substituting the values into the equation, we can solve for u:
0 = u * 3 s + (1/2)(-9.8 m/s²)(3 s)^2
0 = 3u - 44.1
3u = 44.1
u = 44.1 / 3
u ≈ 14.7 m/s
Therefore, the initial velocity of the ball is approximately 14.7 m/s upwards.