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 Question from Jessie, a student: Romeo is chucking pebbles gently up to Juliet's window, and he wants the pebbles to hit the window with only a horizontal component of velocity. He is standing at the edge of a rose garden 4.5m below her window and 5.0m from the base of the wall. How fast are the pebbles going when they hit her window?

Hello Jessie.

Since there is no horizontal acceleration, the horizontal component of velocity when the pebble strikes the window is the same as the horizontal component of velocity when it leaves Romeo's hand, let's call that vh.

To find this constant horizontal velocity, we need the horizontal distance (given) and the time between the pebble leaving Romeo's hand and the window strike.

We know that the vertical speed of the pebble when it hits the window must be zero because he is going to throw the pebble such that it has only a horizontal component of velocity when it strikes the window.

This means that the time it takes gravity to slow the vertical component of velocity to zero is the same time as it takes to move horizontally from Romeo's hand to the window.

The formula that relates two velocities, one distance and one acceleration together for constant linearly accelerated objects is one you should have encountered already:

v2 = v0 2 + 2adv

 where: v is the final vertical velocity ( 0 m/s) v0 is the initial vertical velocity (unknown) a is the acceleration due to gravity (-9.8 m/s2 ) dv is the vertical displacement (4.5 m)

Once you have this vertical component of initial velocity, you can use it to find the time by using this formula:

v= v0 + at

This value t is the time you need to use to find the constant horizonal velocity using the fundamental formula:

dh = vh t

where dh represents the horizontal displacement and vh represents the horizontal velocity you want to find.

Hope this helps,
Stephen La Rocque.

Math Central is supported by the University of Regina and The Pacific Institute for the Mathematical Sciences.