Question

A 200 kg weather rocket is loaded with 100 kg of fuel and fired straight up. It accelerates upward at 30 m/s2 for 30 s, then runs out of fuel. Ignore any air resistance effects. What is the rocket’s maximum altitude? How long is the rocket in the air before hitting the ground?

Answer 1

Answer:

The maximum altitude of the rocket is 54827 m.

The rocket is in the air for  227.6 s

Explanation:

Hi there!

The equation of height and velocity of the rocket are the following:

h = h0 + v0 · t + 1/2 · a · t²   (while the rocket still have fuel).

h = h0 + v0 · t + 1/2 · g · t² (when the rocket runs out of fuel).

v = v0 + a · t  (while the rocket still have fuel).

v = v0 + g · t (when the rocket runs out of fuel).

Where:

h = height of the rocket at time t.

h0 = initial height.

v0 = initial velocity.

a = acceleration due to fuel.

t = time.

g = acceleration due to gravity (-9.8 m/s², downward).

v = velocity of the rocket.

First, let´s calculate the height and velocity reached by the rocket before it runs out of fuel:

h = h0 + v0 · t + 1/2 · a · t²

Since the rocket starts from rest and from the ground, then, v0 = 0 and h0 = 0.

h = 1/2 · a · t²

h = 1/2 · 30 m/s² · (30 s)²

h = 13500 m

The velocity reached before the rocket runs out of fuel will be:

v = v0 + a · t   (v0 = 0)

v = a · t

v = 30 m/s² · 30 s

v = 900 m/s

After 30 s, the rocket runs out of fuel but it continues going up with an initial velocity of 900 m/s. At the maximum point, the velocity of the rocket is equal to zero. Then, using the equation of velocity we can find the time at which the rocket is at its maximum height:

v = v0 + g · t

In this case, the initial velocity is the velocity reached while the rocket still had fuel: 900 m/s. At the maximum height, v = 0:

v = 0

0 = v0 + g · t

0 = 900 m/s - 9.8 m/s² · t

-900 m/s / -9.8 m/s² = t

t = 91.8 s

The maximum heght will be:

h = h0 + v0 · t + 1/2 · g · t²

h = 13500 m + 900 m/s · 91.8 s - 1/2 · 9.8 m/s² · (91.8 s)²

h = 54827 m

The maximum altitude of the rocket is 54827 m.

From this height, the rocket starts to fall. Let´s find the time at which the height is zero (the rocket hits the ground):

h = h0 + v0 · t + 1/2 · g · t²    (v0 = 0 because the rocket falls from the maximum height at which the velocity is zero).

0 = 54827 m - 1/2 · 9.8 m/s² · t²

Solving for t:

√(-54827 m / -4.9 m/s²) = t

t = 105.8 s

Then, the total time of flight is:

30 s + 91.8 s + 105.8 s = 227.6 s

The rocket is in the air for  227.6 s