A student stands at the edge of a cliff and throws a stone horizontally over the edge with a speed of 18.0 m/s. The cliff is 50.0m above a flat, horizontal beach. With what angle of impact does the stone land (in degrees).
v = Vo -9.8 t
z = h + Vo t - 4.9 t^2
here Vo = initial speed up = 0
h = 50
z = final height = 0
so
0 = 50 - 4.9 t^2
so t = 3.19 seconds to fall
v = 0 - 9.8 * 3.19 = -31.3 m/s
so the speed down is 31.3 and the horizontal speed is still 18
tangent of angle from vertical = 18/31.3
To find the angle of impact at which the stone lands, we need to use the concept of projectile motion. Here's how you can solve this problem:
Step 1: Identify the given values:
- Initial horizontal velocity (horizontal component of the stone's velocity) = 18.0 m/s
- Vertical displacement (height of the cliff) = 50.0 m
- Acceleration due to gravity (vertical component of acceleration) = 9.8 m/s^2
Step 2: Find the time of flight:
Since the stone is thrown horizontally, the initial vertical velocity is zero. We can use the equation of motion for vertical displacement: s = ut + (1/2)at^2. Here, s represents the vertical displacement, u represents the initial vertical velocity, a represents the acceleration, and t represents time.
Given:
- s = -50.0 m (negative because the displacement is downward)
- u = 0 m/s
- a = 9.8 m/s^2
Using the equation, we can rearrange it to solve for t:
- s = ut + (1/2)at^2
- -50 = 0*t + (1/2)(9.8)t^2
- -50 = (4.9)t^2
- -50/4.9 = t^2
- t ≈ √10.2041
- t ≈ 3.19 seconds (approx.)
Step 3: Find the horizontal displacement:
To find the horizontal displacement, we can use the equation: s = ut, where s represents the horizontal displacement, u represents the initial horizontal velocity, and t represents time.
Given:
- u = 18.0 m/s
- t = 3.19 s
Using the equation, we can find the horizontal displacement:
- s = ut
- s = 18.0 m/s * 3.19 s
- s ≈ 57.42 m (approx.)
Step 4: Find the angle of impact:
The angle of impact can be found using the tangent function. The tangent of an angle is equal to the ratio of the vertical displacement to the horizontal displacement.
Given:
- Vertical displacement = -50.0 m
- Horizontal displacement = 57.42 m
Using the equation, we can calculate the angle of impact:
- tanθ = Vertical displacement / Horizontal displacement
- tanθ = -50.0 m / 57.42 m
- tanθ ≈ -0.87
To find the angle, we need to take the inverse tangent (also known as arctan):
- θ ≈ arctan(-0.87)
- θ ≈ -41.85 degrees
Therefore, the stone lands at an angle of approximately -41.85 degrees, where the negative sign indicates that it is below the horizontal.