Mix an inspirational T-shirt from NASA with some concepts from projectile motion and work and energy, plus a wee bit of calculus, and learn how to derive the expression for escape velocity. Join Romeo on his quest to rid Earth of his physics textbook once and for all.
Escape velocity is the minimum speed for a projectile to escape a massive object’s gravitational field. Calling it velocity is actually a misnomer because it doesn’t have direction; it should actually be “escape speed” but I have to admit that it just sounds better to call it escape velocity.
Escape velocity is cool! You’ll need it if you ever want to send something far, far away, never to return. For example, Romeo secretly plans to launch his physics textbook permanently away from his home planet Earth. The problem is, Romeo has seen the formula but doesn’t really understand it. Luckily it’s fairly easy to derive from first principles.
Yay! Let’s dive in.
Like many people, Romeo spends most of his time on Earth’s surface, and this is where he plans to launch his book; he just needs to know how fast to throw it. Romeo has tried before, and every time the book left his hands, gravity pulled it back down. Even when Romeo threw the book partway to the moon, Earth’s gravitational field brought it back home, so Romeo wants to be able to throw it infinitely far away.
Looking at a point in between the surface and infinitely far away, we see the force of gravity pulling the book towards Earth. Gravity is the only force acting on the book, and to find the escape velocity we can consider the work done by gravity as the book moves from Earth’s surface to infinitely far away. The work done by a constant force is simply the force magnitude times the displacement times cosine of the angle between the force and the displacement. In this case the angle is always 180 degrees and cosine of 180 degrees is negative 1. Unfortunately the force of gravity is not constant; it depends on the distance from Earth’s center according to the formula F_g = GMm/s^2, where G is the gravitational constant, big M is Earth’s mass, little m is the book’s mass, s is the distance from Earth’s center to the book, and F_g is the magnitude of the force of gravity between Earth and the book. Since F_g varies with s, we need to set up an integral from s = r at Earth’s surface to s = infinite when the book is infinitely far away.
Evaluating this integral is fairly easy. G, big M, and little m are constants, as is cosine of 180 degrees, and I bring them out of the integral, which leaves only 1 over s^2 ds. I can integrate this, getting negative 1/s, and then evaluate to find that the total work done by gravity, when the book moves from r to infinitely far away, is equal to negative GMm/r.
Then I use the work-energy theorem, W_net = delta K, to relate the work done by gravity to the book’s speed. The only force acting on the book is gravity, so the net work – or total work – is just W_g: GMm/r. Delta K is K_final – K_initial. Expanding these terms gives GMm/r = 1/2 mv_f^2 – 1/2 mv_i^2. v_f is zero because we want to find the minimum v_i for the book to completely escape Earth’s gravitational field, and that corresponds to giving the book just as much initial kinetic energy as the work that gravity will do to slow the book down while it’s hurtling away. The book’s mass cancels out, as do the negative signs on both sides, so I just multiply by 2 and then take the square root to find that v_i is the square root of 2Gm/r.
So the escape velocity depends on the universal gravitational constant, the mass of whichever massive object (ahem: planet) you’re escaping from, and the distance to that object’s center. Now Romeo has some “work” to do… Ha ha ha…
I’m Scott Redmond and I help students pass physics. Visit http://ift.tt/1QgRKat for details.
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