Need help physics question on projectile motion?

Question

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 8.00 meters below her window and 11.0 m from the base of the wall. How fast are the pebbles going when they hit her window?

Answer 1

the point here is that juliet's window must be at the apex of motion; at that point, there is only horizontal motion and no vertical motion

the data tell us that the apex of motion is at 8 meters; this means the initial vertical speed must be such that it drops to zero at 8 meters

we can find the initial speed from :

vf^2=v0^2 +2ad where vf and v0 are final and initial speeds, a is accel (here = -9.8 m/s/s) and d is distance, here = 8 m.

Notice that we use the vertical components since we are interested in the vertical component of motion.  Using these values, we find that the intial vertical velocity is:

v0^2=2gd or that v0=12.5 m/s

with an acceleration of 9.8 m/s/s, it will take the rock a time of:

t=12.5 m/s / 9.8 m/s/s = 1.3 s to reach the top of its motion (you get the same answer by finding the time it takes to fall 8 m, namely dist=1/2 g t^2)

So, the rock is in the air for 1.3 s before it reaches its apex, at which point it must strike the window which is 11 m horizontally away from his throwing position.  This means the pebbles must cover 11m in 1.3 s or have a horizontal speed of 8.6 m/s.  Since the horizontal speed does not vary (excluding air friction) this is the speed with which the rock will hit the window.  This speed, incidentally, is 19 mi/hr...will Juliet's window survive the impact?

This analysis assumes that the vertical distance between the launching point and Juliet's window is 8 m.

Answer 2

First figure out how fast the vertical component needs to be.

The time it takes to go up is that same as dropping a pebble from that height, so use d= 1/2at^2 to solve for time t.

With the time, you know how fast the horizontal component needs to be to go the 11 meters in time t.