A student stands at the edge of a cliff and throws a stone horizontally over the edge with a speed of 21.0 m/s. The cliff is h = 60.0 m above a flat horizontal beach.
A)How long after being released does the stone strike the beach below the cliff?
B)With what speed and angle of impact does the stone land?
how long does it take to fall 60m?
h=1/2 g t^2
vf=g t
then, for the speed, combine horizontal speed with the vertical speed as vectors.
To answer this question, you can use basic kinematics equations for projectile motion. Here's how you can approach it:
A) How long after being released does the stone strike the beach below the cliff?
1. Identify the known variables:
- Initial horizontal velocity (Vx): 21.0 m/s
- Initial vertical velocity (Vy): 0 m/s (since the stone is thrown horizontally)
- Height of the cliff (h): 60.0 m
- Acceleration due to gravity (g): 9.8 m/s^2
2. Find the time it takes for the stone to fall from the cliff to the beach using the vertical motion equation:
- The equation to use is: h = (1/2)gt^2
- Rearrange the equation to solve for time (t): t = sqrt(2h / g)
Substituting the given values, we get:
t = sqrt(2 * 60.0 / 9.8) ≈ 3.19 seconds
Thus, it takes approximately 3.19 seconds for the stone to strike the beach below the cliff.
B) With what speed and angle of impact does the stone land?
1. Calculate the horizontal distance traveled by the stone using the horizontal motion equation:
- The formula for distance (d) can be written as: d = Vx * t
Substituting the values we already know, we get:
d = 21.0 * 3.19 ≈ 67.0 meters
The stone lands at a horizontal distance of approximately 67.0 meters from the edge of the cliff.
2. Calculate the vertical component of the stone's final velocity using the vertical motion equation:
- The equation to use is: Vy = gt
- Since the stone falls vertically downward, the final velocity in the vertical direction (Vy) is equal to -g*t (negative sign indicates downward direction)
Substituting the value of time, we get:
Vy = -9.8 * 3.19 ≈ -31.3 m/s
3. Calculate the magnitude of the stone's final velocity using the Pythagorean theorem:
- The magnitude of the velocity (V) can be calculated from its horizontal (Vx) and vertical (Vy) components as follows: V = sqrt(Vx^2 + Vy^2)
Substituting the given values, we have:
V = sqrt(21.0^2 + (-31.3)^2) ≈ 38.1 m/s
Thus, the stone lands with a speed of approximately 38.1 m/s.
4. Calculate the angle of impact using the inverse tangent function:
- The angle (θ) can be calculated using the equation: θ = atan(Vy / Vx)
Substituting the given values, we get:
θ = atan((-31.3) / 21.0) ≈ -56.5°
The angle of impact is approximately -56.5°. Since the stone is thrown horizontally, the angle is negative (measured below the horizontal).
Therefore, the stone lands with a speed of approximately 38.1 m/s at an angle of approximately -56.5°.