Question

1. During a medieval siege of a castle, the attacking army uses a trebuchet to hurl heavy stones at the castle walls. If the trebuchet launches the stones with a velocity of +35.0 m/s at an angle of 60.0°, how long does it take the stone to hit the ground? What is the maximum distance that the trebuchet can be from the castle wall to be in range? How high will the stones go? Show all your work.

Answer 1

The first thing you need to figure out is the vertical component of the stone's velocity. This is the velocity that gravity will be working against--giving you the necessary means to determine the time of the stone's flight.

The vertical component, (Vy), of the velocity is found using simple trig. V(y) = 30 m/s x sin(50)
Therefore, V(y) = 22.981 m/s

Now, you need to find the time it will take for gravity to reduce this upwards velocity to zero, and then double it (so you can account for the time it takes to fall back to the ground).

So... V(f) = V(i) + a x t (Where V(f) is final velocity, V(i) is initial velocity, a is acceleration, and t is time)

0 m/s = 22.981 m/s + -9.8 m/s^2 x t => t = 2.345 sec

REMEMBER--this is the time the rock takes to reach its apex. Double this time to get a final answer of 4.690 sec.

Now, since we know that the rock will be traveling in the air for 7.817 seconds, we can find the total horizontal distance it will travel by multiplying this figure with the horizontal component of its velocity.
This can be found using trig as well. V(x) = 30 m/s x cos(50). Therefore, the horizontal velocity is 19.284 m/s.

Now... 19.284 m/s x 4.690 sec = 90.440 meters.

To find the highest point of a stone's trajectory, use the first time value we solved for (remember, it was 2.345 seconds). This is also the time that it would take for the stone to fall from its maximum height (because this is the time at the apex).

So use the formula D = 0.5 x a x t^2 (Where D = distance, a = acceleration, and t = time)

Plug in the values... D = 0.5 x 9.8 m/s^2 x 2.345^2 => D = 26.945 meters

So all of your final answers will be:

TIME TO HIT GROUND: 4.690 seconds
MAXIMUM RANGE: 90.440 meters
MAXIMUM HEIGHT: 26.945 meters

I hope this helps!