Multiple Choice
You launch a rocket with an initial speed of m/s from Earth's surface. At what height above the Earth will it have ¼ of its initial launch speed? Assume the rocket's engines shut off after launch.
8. Centripetal Forces & Gravitation
Gravitational Potential Energy
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How much energy is required to move a 1000-kg object from Earth's surface to a height twice Earth's radius?
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Verified step by step guidance1
Identify the gravitational potential energy formula for an object at a distance from the center of the Earth: \( U = -\frac{G M m}{r} \), where \( G \) is the gravitational constant, \( M \) is the mass of the Earth, \( m \) is the mass of the object, and \( r \) is the distance from the center of the Earth.
Determine the initial and final distances from the center of the Earth. The initial distance is Earth's radius \( R \), and the final distance is \( 3R \) (since the object is moved to a height twice Earth's radius).
Calculate the initial gravitational potential energy \( U_i = -\frac{G M m}{R} \) and the final gravitational potential energy \( U_f = -\frac{G M m}{3R} \).
Find the change in gravitational potential energy, which is the energy required to move the object: \( \Delta U = U_f - U_i = -\frac{G M m}{3R} + \frac{G M m}{R} \).
Simplify the expression for \( \Delta U \) to find the energy required: \( \Delta U = \frac{2G M m}{3R} \). Substitute the known values for \( G \), \( M \), \( m \), and \( R \) to calculate the energy.
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