Chapter 4, Problem 68PQ

Frequently, a weapon must be fired at a target that is closer than the weapon’s maximum range. To hit such a target, a weapon has two possible launch angles (Fig. P4.68A): one higher than 45° (θ_H) and one lower than 45° (θ_L). Although the displacement of the projectile is the same for the two angles, a projectile launched at θ_H has a longer flight time and a higher peak position than one launched at θ_L. Usually, some tactical situation makes one angle preferable to the other. For example, if the projectile must go over some nearby object such as a grove of trees, the higher angle may be desirable. A shorter flight time and therefore θ_L are preferable if the target is mobile.

In practice, many weapons are designed to operate either at angles lower than 45° or at angles higher than 45°, but not both. Tanks, for example, often must face mobile targets; to minimize the time the target has to move, tanks fire at low angles. Grenades, on the other hand, are launched at high angles because a soldier launching a grenade is often close to the target, but has no armor plating for protection. The high launch angle allows the soldier to stay out of sight by hiding behind some obstacle, and the longer flight time may make it possible for the soldier to move farther from the exploding grenade.

FIGURE P4.68

Imagine an unusual scenario in which a large gun mounted on a vehicle is required to hit an explosives factory (Fig. P4.68B). A huge explosion is expected, and there must be time for the gunner to retreat. A grove of trees provides cover. The maximum range of the gun is 17.6 km, and the maximum speed of the vehicle is 80.0 km/h. a. What is the muzzle speed v₀? (Muzzle speed is the speed at which the projectile leaves the barrel of the gun.) b. The target is 5.5 km away. Find the low angle θ_L and the high angle θ_H at which the gunner may aim so as to hit the target. c. Find the time the projectile takes to hit the target for both angles. d. Assume the vehicle retreats at its maximum speed (80.0 km/h) to be as far from the ensuing explosion as possible. How far is it from the factory at the time of the explosion for each launch angle?