a boy who is 1.5m tall throws a ball into the air the ball attains a height a of 3m above the boy's head and then falls back to the ground the boy's hand was 1.2 m above the ground when he released the ball

how long does it take to reach it's maximum height?
what is the initial velocity , final velocity off the ball

initial height= 2.7m
max height=4.5m

I would do this with energy. You know the initial potential energy.... mghi, and the final potential energy mg*hmax. The difference represents the kinetic energy of the ball initally 1/2 m v^2

The final kinetic energy is equal to mg*hmax

To find the time it takes for the ball to reach its maximum height, we can use the concept of projectile motion.

First, let's calculate the initial velocity (v0) of the ball when it is released. We know that the initial height (hi) is 2.7m and the final height (hf) is 4.5m.

Using the formula for potential energy, mgh, we can equate the initial potential energy to the final potential energy:

m * g * hi = m * g * hf

Since the mass (m) and the acceleration due to gravity (g) are common to both sides of the equation, they cancel out:

hi = hf

Substituting the values, we have:

2.7m = 4.5m

Now, let's calculate the time it takes for the ball to reach its maximum height using the equation of motion:

hf = hi + v0t - 0.5 * g * t^2

Since the ball reaches its maximum height when its velocity becomes zero (vf = 0), we can simplify the equation:

0 = v0 - g * t

Solving for time (t), we get:

t = v0 / g

Now, let's calculate the initial velocity (v0) of the ball using the equation:

vf^2 = v0^2 - 2g * (hf - hi)

Since vf is zero when the ball reaches its maximum height, we can simplify the equation:

0 = v0^2 - 2g * (hf - hi)

Solving for v0, we have:

v0 = sqrt(2g * (hf - hi))

Substituting the values, we have:

v0 = sqrt( 2 * 9.8 * (4.5 - 2.7) )

Therefore, the initial velocity of the ball is approximately 6.26 m/s.

To summarize:

- The time it takes for the ball to reach its maximum height is t = v0 / g.
- The initial velocity of the ball is v0 = sqrt(2g * (hf - hi)).

Using these formulas and the given values, you can now calculate the time and initial velocity of the ball.