A knight sits on a castle wall during a siege. To while away the time, he notes that boul- ders catapulted from below land on the top of his wall with a vertical velocity of 8.3 m/s. If he is 41 m above the catapult, what is the initial velocity of the boulders? The acceleration of gravity is 9.8 m/s2 . Answer in units of m/s.
To find the initial velocity of the boulders, we can use the concept of projectile motion. In this case, we are given the vertical velocity of the boulders (8.3 m/s) and the height of the knight's position (41 m). We can use the equations of motion to solve for the initial velocity.
The equation we'll use is:
vf^2 = vi^2 + 2 * a * d
Where:
vf is the final velocity (which is 0 as the boulders reach their maximum height),
vi is the initial velocity (which we're trying to find),
a is the acceleration due to gravity (-9.8 m/s^2),
and d is the displacement in the vertical direction (41 m).
Plugging in the given values:
0 = vi^2 + 2 * (-9.8 m/s^2) * 41 m
Simplifying the equation:
0 = vi^2 - 2 * 9.8 m/s^2 * 41 m
0 = vi^2 - 80.4 m^2/s^2
Rearranging the equation to solve for vi:
vi^2 = 80.4 m^2/s^2
Taking the square root of both sides:
vi ≈ ± 8.97 m/s
Since we are looking for the initial velocity, which is a magnitude, we can ignore the negative sign.
Therefore, the initial velocity of the boulders is approximately 8.97 m/s.