Question

you are on the street trying to hit a friend in his dorm window with a water balloon. he has a similar idea and is aiming at you with his water balloon. you aim straight at each other and throw at the same instant. Do the water ballons hit each other? explain why or why not

Answer 1

Here it is given that you and your friend both are aiming at each other

So here we can find the relative motion of two balls

first we will write the components of water balloon of your hand

$v_x = v cos\theta$

$v_y = v sin\theta$

$a = -g$

similarly now for your friends balloon components are given as

$v_x' = - v'cos\theta$

$v_y' = - v' sin\theta$

$a = -g$

now to find the relative motion

$v_{rx} = (v + v') cos\theta$

$v_{ry} = (v + v') sin\theta$

$a_r = 0$

so here the relative motion of two balls is always in a straight line with respect to each other

$tan\phi = \frac{v_{ry}}{v_{rx}}$

$tan\phi = tan\theta$

so they move in straight in with respect to each other and hit directly

so in this case we can say that two balloons always hit each other when they are targeted towards each other because the motion is always towards each other and the time to hit each other is given by

$t = \frac{d}{\sqrt{v^2+v'^2}}$

here d is the straight distance between you and your friend balloon

also here net speed is

$v = \sqrt{v^2 + v'^2}$