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

To find the rate at which the gravitational force is changing, we need to differentiate the given expression for the gravitational force with respect to time. Let's denote the rate of change of gravitational force by dF/dt.

Given:
F = k/r^2
k = 10^13 newton • km^2
r = 10^4 km
dr/dt = 0.2 km/sec (rate of change of distance)

To find dF/dt, we need to differentiate both sides of the equation F = k/r^2 with respect to time.

Differentiating F = k/r^2 with respect to time gives us:

dF/dt = d(k/r^2)/dt

Now, let's differentiate k/r^2 using the quotient rule:

dF/dt = (d(k)/dt • r^2 - k • d(r^2)/dt) / (r^2)^2

To differentiate k with respect to time, we assume k is a constant. Therefore, d(k)/dt is 0.

dF/dt = (-k • d(r^2)/dt) / (r^2)^2

Next, let's differentiate r^2 with respect to time:

d(r^2)/dt = 2r • (dr/dt)

Now, plug in the given values:

k = 10^13 newton • km^2
r = 10^4 km
dr/dt = 0.2 km/sec

dF/dt = (-10^13 newton • km^2 • 2(10^4 km) • 0.2 km/sec) / (10^4 km)^2)^2

Simplifying further:

dF/dt = (-4 • 10^17 newton • km^3 • km/sec) / (10^8 km^2)^2

dF/dt = (-4 • 10^17 newton • km^3 • km/sec) / 10^16 km^4

dF/dt = -4 • 10 newton • km/sec (after canceling units and simplification)

Therefore, the rate at which the gravitational force is changing is -4 • 10 newton • km/sec.

No, the answer is not 2,000 Newtons. Let's calculate the rate at which the gravitational force is changing at that moment.

To find the rate of change, we need to differentiate the given gravitational force equation with respect to time. So, let's start by differentiating the equation:

F = k / r^2

Using the power rule for differentiation, we get:

dF/dt = d(k / r^2) / dt

To differentiate k with respect to t, we assume it is a constant, so its derivative is zero.

d(k / r^2) / dt = (0 - k * d(r^2) / dt) / r^4

Next, let's differentiate the distance equation with respect to time:

r = 10^4 km

Taking the derivative using the chain rule, we have:

dr / dt = d(10^4 km) / dt

Since the rocket is moving away, the rocket's distance from the center of the Earth is increasing; thus, dr / dt is equal to the given velocity:

dr / dt = 0.2 km/sec

Now, let's substitute these values back into the equation:

dF/dt = - k * (2r * dr / dt) / r^4 (note: 2r * dr / dt is the derivative of r^2)

Plugging in the values:

dF/dt = - (10^13 Newton • km^2) * (2 * 10^4 km * 0.2 km/sec) / (10^4 km)^4

Simplifying this equation will give us the rate of change of the gravitational force.