THE ZOOKEEPER AND THE MONKEY
A monkey escapes from the zoo and climbs a tree. After failing to entice the monkey down, the zookeeper fires a tranquilizer dart directly at the monkey (Fig. 3.26).
The monkey lets go at the instant the dart leaves the gun. Show that the dart will always hit the monkey before he hits the ground and runs away.
IDENTIFY and SET UP:
We have two bodies in projectile motion:
the dart and the monkey. They have different initial positions and initial velocities, but they go into projectile motion at the same time t = 0. We’ll first use Eq. (3.19) (x=\left(v_{0} \cos \alpha_{0}\right) t) to find an expression for the time t when the x-coordinates x_{\text {monkey }} \text { and } x_{\text {dart }} are equal. Then we’ll use that expression in Eq. (3.20) (y=\left(v_{0} \sin \alpha_{0}\right) t-\frac{1}{2} g t^{2}) to see whether y_{\text {monkey }} and y_{\text {dart }} are also equal at this time; if they are, the dart hits the monkey. We make the usual choice for the x- and y-directions, and place the origin of coordinates at the muzzle of the tranquilizer gun (Fig. 3.26).
EXECUTE:
The monkey drops straight down, so x_{\text {monkey }} = d at all times. From Eq. (3.19), x_{\text {dart }}=\left(v_{0} \cos \alpha_{0}\right) t. We solve for the time t when these x-coordinates are equal:
d=\left(v_{0} \cos \alpha_{0}\right) t \quad \text { so } \quad t=\frac{d}{v_{0} \cos \alpha_{0}}We must now show that y_{\text {monkey }}=y_{\text {dart }} at this time. The monkey is in one-dimensional free fall; its position at any time is given by Eq. (2.12) (x=x_{0}^{x}+v_{0 x} t+\frac{1}{2} a_{x} t^{2}), with appropriate symbol changes. Figure 3.26 shows that the monkey’s initial height above the dart-gun’s muzzle is y_{\text {monkey }-0}=d \tan \alpha_{0}, \text { so }
y_{\text {monkey }}=d \tan \alpha_{0}-\frac{1}{2} g t^{2}From Eq. (3.20) (y=\left(v_{0} \sin \alpha_{0}\right) t-\frac{1}{2} g t^{2}),
y_{\mathrm{dart}}=\left(v_{0} \sin \alpha_{0}\right) t-\frac{1}{2} g t^{2}Comparing these two equations, we see that we’ll have y_{\text {monkey }}=y_{\text {dart }} \text { (and a hit) if } d \tan \alpha_{0}=\left(v_{0} \sin \alpha_{0}\right) t when the two
x-coordinates are equal. To show that this happens, we replace t with d /\left(v_{0} \cos \alpha_{0}\right), the time when x_{\text {monkey }}=x_{\text {dart }}. Sure enough,
EVALUATE: We’ve proved that the y-coordinates of the dart and the monkey are equal at the same time that their x-coordinates are equal; a dart aimed at the monkey always hits it, no matter what v_0 is (provided the monkey doesn’t hit the ground first). This result is independent of the value of g, the acceleration due to gravity. With no gravity (g = 0), the monkey would remain motionless, and the dart would travel in a straight line to hit him. With gravity, both fall the same distance gt^2/2 below their t = 0 positions, and the dart still hits the monkey (Fig. 3.26).
