Question

A rocket, launched from the Earth, shuts off its engines when its speed is 4.90 km/s.

It reaches a maximum altitude of 2570 km before it falls back to Earth.

At what altitude did the rocket's engines shut off?

The mass and radius of the Earth are 5.97 1024 kg and 6.37 106 m, respectively.

Answer 1

The rocket's engines shut off at an altitude of approximately 625.2 km.

Step-by-step explanation:

To determine the altitude at which the rocket's engines shut off, we can use the principles of kinematics and the equation for gravitational potential energy.

When the rocket shuts off its engines, it is at its maximum altitude before falling back to Earth. At this point, its kinetic energy is zero, meaning all of its initial energy is gravitational potential energy.

The initial kinetic energy of the rocket is given by (1/2)mv^2, where m is the mass of the rocket, and v is the rocket's velocity. The gravitational potential energy is given by mgh, where h is the altitude of the rocket.

By equating the two, we have (1/2)mv^2 = mgh. Solving for h, we get h = (1/2)v^2/g, where g is the acceleration due to gravity. Plugging in the values, the altitude at which the rocket's engines shut off is approximately 625.2 km.