Question

9, Galileo throws a rock from the top of the Leaning Tower of Pisa at an upward angle of 69.00 with speed vo. The rock is in flight for 5.2 s and hits the ground 15 m from the base of the building. Ignore air resistance and ignore the fact that the two tilts a bit. a) What is the speed vo? Vo b) How high off the ground is the top of the tower? Height of tower c) What is the speed of the rock just before it hits the ground? Final Speed of rock

Answer 1

Given,

$\theta$ = 69 degrees

t = 5.2 sec ; r = 15m

(a) r = Vo² sin(2 $\theta$ ) / g

Vo² = r g / sin(2 $\theta$ ) = 15 x 9.8 / sin (2x69) = 147/0.669 = 219.73

Vo = sqrt(219.73) = 14.82 m/s

Hence, Vo = 14.82 m/s

(b)Let the max height reached be H

Now the y comp of velocity will be

Voy = Vo sin( $\theta$ ) = 14.82 x sin69 = 13.83 m/s

Now the time taken by the rock to reach the max height will be given by

t' = Voy / g = 13.83/9.8 = 1.41 sec

From eqn of motion we have, s = ut + 1/2gt²,

h = 1/2 g t'² = 1/2 x 9.8 x (1.41)² = 9.74 meters

again we are provided with t = 5.2 sec

time for the rock to fall freely from max height, t'' = 5.2 - 1.41 = 3.8 s

h '= 1/2 g t''² = 0.5 x 9.8 x (3.8)² = 70.8 m

Hence, the max height H = h' - h = 70.8 - 9.74 = 61.06 m

Hence, H = 61.1 meters

c)Vertical speed of the rock just before hitting the groung will be

Vy = sqrt(2 g H) = sqrt (2 x 9.8 x 61.1) = 34.6 m/s

Horizontal speed, Vox = Vo x cos( $\theta$ ) = 14.82 x cos(69) = 5.3 m/s

So the final speed of rock = sqrt [ (34.6)² + (5.3)²] = 35 m/s

Hence, final speed = V = 35m/s