Question

two boys throw balls A and B from the window with a speed v(0) and kv(0), respectively, where k is a constant. Show that the balls will collide if k = cos(theta2) / cos(theta1).

Answer 1

Given:

Two boys throw balls A and B from the window with a speed v(0) and kv(0) respectively, where k is a constant.

To show:

$k = \dfrac{ \cos( \theta2) }{ \cos( \theta1) }$

Calculation:

Boy A threw the ball at a velocity v_(0) at an angle of $\cos(\theta2)$

So, velocity of the ball at the highest point:

$v = v_{0} \cos( \theta2)$

Boy B threw the ball at a velocity kv_(0) at an angle of $\cos(\theta1)$.

So, velocity of the ball at the highest point:

$v = kv_{0} \cos( \theta1)$

The balls will collide if the the velocity vectors at the highest point is equal and opposite;

$\therefore \: v_{0} \cos( \theta2) = kv_{0} \cos( \theta1)$

$= > k = \dfrac{ \cos( \theta2) }{ \cos( \theta1) }$

[Hence proved]