Problem Description:

A bomb is dropped from a balloon hovering at an altitude of 800 ft. A gun emplacement is located on the ground directly below the balloon. The gun fires a projectile straight upward toward the bomb exactly 2 seconds after the bomb is released. With what initial speed should the projectile be fired in order to intercept the bomb at an altitude of 400 ft.?

Reference: This problem comes from Prob. 23 on page 17 of the Differential Equations text by Edwards and Penny (Prentice Hall Inc., 1996).

Problem Solution:

Bomb Analysis

A force balance on the bomb gives

This equation is separable and easily integrated to give

Applying the initial condition that gives . Therefore, v_b(t) is simply

Now considering the location of the bomb versus time gives

To determine the integration constant, we use the fact that , so c = 800 ft and

From the problem statement, we want to know how long it takes for the bomb to reach 400 ft.

Therefore,

With g = 32.2 ft/s², the time of impact is t = t_f = 4.984 seconds.

Projectile Analysis

Performing a similar analysis on the projectile gives a force balance,

with solution

In this case the initial condition is written as, which gives , and

Now the position of the projectile is given by the solution of

which gives

The projectile is on the ground at t = 2 seconds, which gives

or . Thus, the projectile position versus time is given as

and, for y_p(t_f) = 400 ft and t = t_f = 4.984 s, we can solve for v_p0, giving

or

A Slight Generalization

One can generalize the relationship between impact time and height and the required initial projectile velocity to achieve the desired impact parameters. For example, if we equate the expressions for y_b(t) and y_p(t) at t = t_f, we get

and solving this for the initial projectile velocity, v_p0, gives