When the mass m of a body is changing with time, Newton’s second law of motion becomes

where F is the net force acting on the body and mv is its momentum. Use (17) in Problems 21 and 22.

In Problem 21, the mass m(t) is the sum of three different masses: where mp is the constant mass of the payload, mv is the constant mass of the vehicle, and mf (t) is the variable amount of fuel.

(a) Show that the rate at which the total mass m(t) of the rocket changes is the same as the rate at which the mass mf (t) of the fuel changes.

(b) If the rocket consumes its fuel at a constant rate , find m(t). Then rewrite the differential equation Problem 21 in terms of and the initial total mass m(0) = m0.

(c) Under the assumption in part (b), show that the burnout time tb >0 of the rocket, or the time at which all the fuel is consumed, is where mf (0) is the initial mass of the fuel.

(reference problem 21)

When the mass m of a body is changing with time, Newton’s second law of motion becomes

where F is the net force acting on the body and mv is it momentum. Use (17) in Problems 21 and 22.

A small single-stage rocket is launched vertically as shown in Figure 1.3.19. Once launched, the rocket consumes its fuel, and so its total mass m(t) varies with time t > 0. If it is assumed that the positive direction is upward, air resistance is proportional to the instantaneous velocity v of the rocket, and R is the upward thrust or force generated by the propulsion system, then construct a mathematical model for the velocity v(t) of the rocket. [Hint: See (14) in Section 1.3.]

(reference problem 14)

The right-circular conical tank shown in Figure 1.3.13 loses water out of a circular hole at its bottom. Determine a differential equation for the height of the water h at time t > 0. The radius of the hole is 2 in., g = 32 ft/s2, and the friction/contraction factor introduced in Problem 13 is c = 0.6.

(reference problem 13)

Suppose water is leaking from a tank through a circular hole of area Ah at its bottom. When water leaks through a hole, friction and contraction of the stream near the hole reduce the volume of water leaving the tank per second to is an empirical constant. Determine a differential equation for the height h of water at time t for the cubical tank shown in Figure 1.3.12. The radius of the hole is 2 in., and g =32 ft/s2.

Solution

Step 1

In this problem, we have to show that the total mass of the rocket changes is the same as the rate at which the mass of the fuel changes.