The gravitational force, F, on a rocket at a distance, r, from the center of the earth is given by F=k/r^2

where k = 10^13 newton • km2. When the rocket is 10^4 km from the center of the earth, it is moving away at 0.2 km/sec. How fast is the gravitational force changing at that moment? Give units. (A newton is a unit of force.)

Is the answer 2,000 Newtons

dF/dr= d/dr (kr^-2)=-2kr^-3

Is the answer then -4 x 10^-13

To find the rate of change of the gravitational force at a given distance, we need to differentiate the equation with respect to time.

Given equation: F = k/r^2

Differentiating both sides with respect to time (denoted as t):
(dF/dt) = d/dt (k/r^2)

Now, let's substitute the given values:
k = 10^13 N • km^2
r = 10^4 km

The equation becomes:
(dF/dt) = d/dt (10^13 / (10^4)^2)

Before differentiating, let's simplify the equation:
(dF/dt) = d/dt (10^13 / 10^8)

To differentiate, we treat k as a constant, and differentiate the variable part with respect to time t.
(dF/dt) = 0 (since the expression is a constant.)

Therefore, the rate of change of the gravitational force at that moment is zero. The units remain unchanged, so the answer is 0 Newtons.