A model rocket is fired straight up from the top of a 45-m-tall building. The rocket has only enough fuel to burn for 4.0 s. But while the rocket engine is burning fuel, it produces an upward acceleration of 55 m/s2. After the fuel supply is exhausted, the rocket is in free fall and just misses the edge of the building as it falls back to the ground. Ignoring air resistance, calculate (a) the height above the ground and the velocity of the rocket when its fuel runs out; (b) the maximum height of the rocket; (c) the time the rocket is in the air; and (d) the rocket's velocity the moment before it hits the ground.

a)y = 485 m ,  v = 220 m / s , b)  y = 2954.39 m , c)   t_total = 51 s ,

d) v = 240.59 m / s

Explanation:

a) We can use vertical launch ratios for this exercise

the speed of the rocket the run out the fuel is

        v = v₀ + a t

the rocket departs with initial velocity v₀ = 0

        v = a t

        v = 55 4

        v = 220 m / s

the height at this point is

        y = y₀ + v₀t + ½ a t²

        y = y₀ + 1/2 a t²

        y = 45 + ½ 55 4²

        y = 485 m

b) the maximum height of the rocket is when its speed is zero

for this part we will use as the initial speed the speed at the end of the fuel (v₀´ = 220 m / s) and the height of y₀´ = 485 m

        v² = v₀´² + 2 g (y-y₀´ )

         0 = v₀´² + 2 g (y-y₀´ )

         y = y₀´ + v₀´² / 2g

         y = 485 + 220 2/2 9.8

         y = 2954.39 m

c) the time that the rocket is in the air is the acceleration time t₁ = 4 s, plus the rise time (t₂) plus the time to reach the ground (t₃)

let's calculate the rise time

           v = v₀´- g t

           v = 0

            t₂ = v₀´ / g

            t₂ = 220 / 9.8

            t₂ = 22.45 s

Now let's calculate the time it takes to get from this point (y₀´´ = 2954.39 m) to the floor

           y = y₀´´ + v₀´´ t - ½ g t²

           0 = y₀´´ - ½ g t²

          t = √ (2 y₀´´ / g)

          t = √ (2 2954.39 / 9.8)

          t = 24.55 s

the total flight time is

       t_total = t₁ + t₂ + t₃

       t_total = 4 + 22.45 + 24.55

        t_total = 51 s

d) the veloicda right now

       v = vo + g t

       v = 9.8 24.55

       v = 240.59 m / s