A rocket is launched straight up from the earth's surface at a speed of 15,000 m/s. What is its speed when it is very far away from the earth?

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Welcome back, everyone. We are making observations about a spacecraft and we are told that the spacecraft is launched from a planet's surface with an initial velocity of 20,000 m per second. Now, the mass of our planet we are told is 5.98 times 10 to the 24th kilograms. And we are told that the radius of our planet is 6.37 times 10 to the 6 m. Now, we are tasked with finding what is the velocity going to be when we are at an infinite distance away from the planet's gravitational yield before getting started here. I do wish to acknowledge our multiple choice answers on the left hand side of our screen. Those are gonna be the values that we strive for. So without further ado let us begin. Well, what can we use here? Well, we're dealing with velocities and masses. The first thing that comes to my mind is the conservation of energy formula. Specifically what we are going to have is that our final kinetic energy plus our final gravitational potential energy is equal to our initial kinetic energy plus our initial gravitational potential energy. So let's go ahead and fill in the terms that we know. What, what does this all mean? Well, we are going to have one half times the mass of our space shuttle times our final velocity minus the gravitational constant times the mass of our planet times the mass of our space shuttle divided by our final radius squared equal to one half MS V knot squared plus GMP MS divided by R knot squared. And what can we do from here? Well, looking at the second term on the left hand side of our equation, if our denominator is infinity, this term is going to go to zero. So now in order to isolate our desired final velocity variable, I'm going to multiply both sides of our equation by two over MS you'll see on the left hand side of our equation that our values cancel out. And what we get, let me go ahead and change colors here. What we get is that our final velocity is equal to our initial velocity squared plus two times our gravitational constant times the mass of our planet divided by our initial radius squared. What is this going to equal? Well, this is gonna be 20,000 squared minus two times 6.67 times 10 to the negative 11th times 5.98 times 10 to the 24th divided by 6.37 times 10, 2 B six squared. Now, what this is going to give us is that our final velocity is equal to 16,576 0.1 m per second. But we need kilometers per second. So we know that there's 1000 m in every kilometer multiplying straight across. We see that meters cancel out on top and bottom. And what we get for our final answer is 16.6 kilometers per second, which corresponds to our final answer. Choice of a Thank you all so much for watching. I hope this video helped. We will see you all in the next one.

A rocket is launched straight up from the earth's surface at a speed of 15,000 m/s. What is its speed when it is very far away from the earth?

Verified Solution

Key Concepts

Kinetic Energy and Potential Energy

Conservation of Energy

Escape Velocity