Can you explain the concept of energy conservation?

Apply the concept of conservation of mechanical energy in order to calculate escape speed from earth's surface?

• The escape speed of an object is the term used for the initial speed needed to escape from the surface of a planet or moon so that, after escape, the object has zero speed left. Apply the concept of conservation of mechanical energy (=GPE+KE) in order to calculate the escape speed from earth's surface. Remember that this is a situation where you cannot use Earth's surface as the "zero level".

• Answer:

  The equation for gravitational potential energy is: GPE = -GMm/r GPE = Gravitational Potential Energy G = Gravitational constant M = Mass of Earth m = Mass of you (or spaceship or whatever bit of matter you want to go at escape velocity) r = distance from Earth As you can see, at r -> infinity, potential energy GPE -> 0. And, as when you are closer than infinity, potential energy is negative. Kinetic energy is the energy of motion: KE = 1/2mv^2 KE = Kinetic energy m = Mass of you (same as above, very important!) v = velocity (what we're looking for) Given conservation of energy: E = GPE + KE = constant E = Total Energy Since you are asking about the exact escape velocity, when GPE = 0 (i.e. when you are at infinity), KE = 0 as well (if KE > 0 you would still have some velocity left, and thus your original velocity would be higher than escape velocity, and KE < 0 is physically meaningless). This means that: E = 0 + 0 = 0 That means that: GPE + KE = 0 or, -GPE = KE That is, your kinetic energy at any point must equal the negative of your potential energy (and since your potential energy is always negative, the kinetic energy must be positive). So, let's sub in the equations from earlier: - (-GMm/r) = 1/2mv^2 we can cancel the m's out and cancel the two negatives on the left side so: GM/r = 1/2v^2 Now, we can directly solve for v: 2GM/r = v^2 v = sqrt(2GM/r) So, what does this mean? This equation says that the escape velocity is can be found by only knowing G (the gravitation constant), M (the mass of Earth, or whatever body you're interested in), and r (the distance you are from its center). Plugging in the numbers: G = 6.674 x 10^-11 M = 5.974 x 10^24 r = 6.378 x 10^6 v = sqrt(2*6.674 x 10^-11*5.974 x 10^24/6.378 x 10^6) = 11,180 m/s (rounded to 4 sig figs...)