One of the great discoveries in science is Newton’s universal law of gravitation, which st...

One of the great discoveries in science is Newton’s universal law of gravitation, which states that the magnitude of the gravitational force exerted by one point mass on an other is proportional to their masses and inversely proportional to the square of the distance between them. Insymbols,

where G is a universal gravitational constant. This constant, first measured by Lord Cavendish in 1798, has acurrently accepted value approximately equal to 6.6726 ×10–11 Nm2/kg2. Newton also showed that the law was valid for two spherical masses. In this case, you assumeth at the mass is concentrated at a point at the center of each sphere.

Suppose that an object with mass m is launched from the earth’s surface with initial velocity v0. Let y represent its position above the earth’s surface, as shown in Figure.

Figure. The object in Exercise.

(a) If air resistance is ignored, use the idea in equation (3.15) to help show that

(3.19)

(b) Assuming that y(0) = 0 (the object is launched from earth’s surface) and v(0) = v0, solve equation (3.19) to show that

(c) Show that the maximum height reached by the object is given by

(d) Show that the initial velocity

is the minimum required for the object to “escape” earth’s gravitational field. Hint: If an object “escapes” earth’s gravitational field, then the maximum height acquired by the object is potentially infinite.