A rocket with a mass of 1000 tons starts vertically but upward under the action of engines emitting gases

A rocket with a mass of 1000 tons starts vertically but upward under the action of engines emitting gases at a speed of 4 km / h, a fuel consumption of 3250 kg / s. With what acceleration the rocket starts its upward movement.

mр = 1000 t = 1,000,000 kg.

mg = 3250 kg.

tg = 1 s.

Vg = 4 km / s = 4000 m / s.

ar -?

but – ?

All data in the problem statement will be converted into units of the international SI system.

Rocket mass: mr = 1000 t = 1,000,000 kg.

1 t = 1000 kg, 1000 t = 1000 * 1000 kg = 1,000,000 kg.

Gas emission velocity: Vg = 4 km / s = 4 * 1000 m / s = 4000 m / s.

1 km = 1000 m.

The force of the emission of gases F
Let’s write Newton’s 2 law for gases that fly out of a rocket nozzle.

F = mg * ar, where

mg is the mass of the gas that is emitted;
ag – acceleration of gases.
Let us write down the definitions for the acceleration of the body: ar = (Vg – V0g) / t, where

Vg is the final velocity of gases;
V0g – initial velocity of gases;
t is the time during which the speed changes.
Since the initial velocity of gases V0g is 0, the formula will take the form: a = Vg / t.

Let us express the force of gas emission from the nozzle: F = mg * Vg / t.

Rocket Takeoff Force
According to Newton’s 3 law, the rocket receives the same force F during takeoff.

We write 2 Newton’s law for a rocket: F = mр * ar, where

mр is the mass of the rocket;
ap is the acceleration of the rocket.
From the ratios mg * Vg / t = mr * ar let us express the acceleration of the rocket during takeoff.

ar = mg * V / t * mr.

Substitute the numerical values ​​from the problem statement into the formula.

ar = 3250 kg * 4000 m / s / 1 s * 1,000,000 kg = 13 m / s ^ 2.

Acceleration ap is acquired by the rocket due to the emission of gases from the nozzle. It is directed vertically upward.

But gravity acts on the rocket, which imparts to the rocket an acceleration of gravity g, which is directed vertically downward.

Therefore, the acceleration of the rocket, a, relative to the surface of the earth, will be determined by the difference: a = ap – g.

a = 13 m / s ^ 2 – 9.8 m / s ^ 2 = 3.2 m / s ^ 2.

Answer: the rocket begins to move with an acceleration of a = 3.2 m / s ^ 2 relative to the earth’s surface.