Name: Motion in a Plane With Constant Acceleration
Uploaded: 2019-07-19
Duration: 6 min 42 s
Description: In this video, we are going to look at the simple concept of motion in a plane with constant acceleration. You'll be able to master the concept in just a few...

Question

A hunter aims his gun and fires a bullet directly at a monkey in a tree. At the instant, the bullet leaves the barrel of the gun, the monkey drops. Will the bullet hit the monkey? Support your answer with proper reasoning.

Accepted Answer

Let the monkey stationed at $A$ in the tree be fired with a gun with a velocity ${\vec{v}}_{0}$ at angle $θ$ with the horizontal along the line of sight as shown in the figure.

If $g$ (the magnitude of the free-fall acceleration) was zero, the bullet would follow the straight-line path $O A$ as shown in the figure and the monkey would float in place of dropping. The bullet would certainly hit the monkey. However, $g$ is not zero, but the bullet still hits the monkey.

The motion of the bullet:

Let ${\vec{v}}_{x 0}$ and ${\vec{v}}_{y 0}$ be the two rectangular components of ${\vec{v}}_{0}$ . Clearly, $v_{x 0} = v_{0} cosθ$ and $v_{y 0} = v_{0} sinθ$
Let the bullet take a time $t$ to reach a point $B (x, y)$ directly below $A$ .
As $O C = x$ and $B C = y$ ,

$x = v_{x 0} t$ and $y = v_{y 0} t + \frac{1}{2} a_{y} t^{2}$

or $y = v_{y 0} t + \frac{1}{2} (- g) t^{2} = v_{y 0} \times \frac{x}{v_{x 0}} - \frac{1}{2} g t^{2}$
or $y = x tanθ - \frac{1}{2} g t^{2} \dots (1)$ $[as \frac{v_{y 0}}{v_{x 0}} = \tan θ]$

Motion of the monkey:
Since monkey is initially not at the origin, let $A C = y_{0}$ . If after time $t$ , monkey's $y$ -coordinate is $y'$ , then
$y' = y_{0} + v_{y 0} t + \frac{1}{2} a_{y} t^{2}$

or $y' = x tanθ - \frac{1}{2} g t^{2} \dots (2)$

(as $tanθ = \frac{A C}{O C} = \frac{y_{0}}{x}, y_{0} = x tanθ$ and $v_{y 0} = 0$ and $a_{y} = g$ )
From equations $(1)$ and $(2)$ , $a_{y} = g$ . Therefore, the bullet will hit the monkey when it reaches $B$ .

The figure shows that during the time of flight of the bullet, both bullet and monkey fall the same distance from their origin $(g = 0)$ .

A hunter aims his gun and fires a bullet directly at a monkey in a tree. At the instant, the bullet leaves the barrel of the gun, the monkey drops. Will the bullet hit the monkey? Support your answer with proper reasoning.

Important Questions on Kinematics II

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