To find the initial speed at which the rock was thrown from the building, calculate the horizontal and vertical components of motion separately, then use the Pythagorean theorem to combine them. The rock was thrown with an initial speed of approximately 32.25 ft/s.
Finding the Initial Speed of the Thrown Rock
To calculate the initial speed with which the rock was thrown, we need to utilize the known information about the motion and apply kinematic equations. The rock is thrown from the edge of a 100-ft tall building and lands 160 ft horizontally from the base after 5.0 s. Keeping air resistance out of the question, the motion of the rock can be divided into two components: horizontal (constant velocity) and vertical (accelerated motion).
First, to find the horizontal speed (ux), we use the formula for constant velocity:
ux = 160 ft / 5.0 s = 32 ft/s.
For the vertical speed (uy), we apply the kinematic equation for distance under constant acceleration:
Here, s is the displacement (100 ft, but it's downward so we take it as negative), uy is the initial vertical speed (which we're solving for), a is the acceleration due to gravity (approximately 32 ft/s2), and t is the time (5.0 s).
By inserting the known values and solving for uy, we find:
-100 ft = uy(5.0 s) + (1/2)(-32 ft/s2)(5.0 s)2
uy = -4 ft/s (upward).
Finally, to determine the initial speed of the throw, we combine the horizontal and vertical components with the Pythagorean theorem:
- Initial Speed = √(ux2 + uy2)
Initial Speed = √((32 ft/s)2 + (-4 ft/s)2) = √(1024 + 16) ft/s
Initial Speed = √(1040) ft/s ≈ 32.25 ft/s
The rock was thrown with an initial speed of approximately 32.25 ft/s.