An object is released from rest when it is a height h above the surface of a planet of mass M and radius R. What is the speed of the object just before striking the surface of the planet? Neglect any air resistance. Let h = 4.0 × 10⁶ m, R = 5.0 × 10⁶ m, and M = 4.0 × 10²⁴ kg. A) 7.8 km/s B) 3.5 km/s C) 5.4 km/s D) 6.9 km/s E) 4.8 km/s

Question

Quizplus · Accepted Answer

The potential energy of the object is converted into kinetic energy as it falls towards the surface of the planet. Using conservation of energy, we can equate the initial potential energy to the final kinetic energy:
mgh = (1/2)mv^2
where m is the mass of the object, g is the acceleration due to gravity, h is the initial height, and v is the final speed.
We can simplify this equation by canceling the mass and solving for v:
v = sqrt(2gh)
The acceleration due to gravity on the surface of a planet is given by:
g = G*M/R^2
where G is the universal gravitational constant.
Substituting the given values for h, M, and R, and solving for v, we get:
v = sqrt(2*G*M/R*(1 - R/(R+h)))
Plugging in the numbers, we get:
v = 6.9 km/s (rounded to one decimal place)
Therefore, the answer is D) 6.9 km/s.

An Object Is Released from Rest When It Is a Height