A person is standing 100 feet from a tall cliff. She throws a rock 80 feet per second at an angle of 45 degrees. Neglecting air resistance and assuming the person is 6 feet tall, does the rock strike the cliff? If so, how far up the cliff does it hit?
To determine whether the rock strikes the cliff and where it hits, we need to analyze its trajectory and compare it to the height of the cliff. Let's break down the problem into steps:
Step 1: Analyze the rock's motion:
When the person throws the rock, it follows a parabolic path. We can analyze its vertical and horizontal components separately.
Vertical Component:
The initial vertical velocity (v_y) of the rock can be calculated using the given angle of 45 degrees and the initial velocity of 80 feet per second.
v_y = v * sin(angle)
v_y = 80 * sin(45)
v_y ≈ 56.57 feet per second
Using kinematic equations, we can determine the time it takes for the rock to reach its peak height (going up) and the time it takes to hit the ground (going down). Since we know that the total time of flight is double the time to reach the peak, we can calculate the total time of flight (t).
t = (2 * v_y) / g
where g is the acceleration due to gravity (approximately 32 feet per second squared).
t = (2 * 56.57) / 32
t ≈ 3.53 seconds
Horizontal Component:
The horizontal distance (d_x) covered by the rock can be calculated using the given initial velocity of 80 feet per second and angle of 45 degrees.
d_x = v * cos(angle)
d_x = 80 * cos(45)
d_x ≈ 56.57 feet per second
Step 2: Determine the maximum height of the rock:
The maximum height reached by the rock can be calculated using the equation:
h_max = (v_y^2) / (2 * g)
h_max = (56.57^2) / (2 * 32)
h_max ≈ 49.9975 feet
Step 3: Analyze the cliff and determine if the rock hits it:
The person is standing 100 feet from the cliff, and the cliff is taller than 49.9975 feet (6 feet taller than the person). Therefore, the rock will hit the cliff.
Step 4: Calculate the distance up the cliff where the rock hits:
To find the distance up the cliff where the rock hits, we need to calculate the horizontal distance covered during the time it takes for the rock to reach the maximum height.
d_up = d_x * t_to_peak
where t_to_peak is the time it takes for the rock to reach the maximum height.
Using the equation d_up = d_x * t_to_peak, we can substitute t_to_peak = t / 2 (as it takes half of the total time to reach the maximum height).
d_up = 56.57 * (3.53 / 2)
d_up ≈ 99.596 feet
Therefore, the rock will strike the cliff approximately 99.596 feet up from its base.
Note: It's important to note that this calculation assumes the ground below the cliff is flat and at the same height as the person's initial position. Also, since this is a simplified calculation neglecting air resistance, it may not exactly match real-world conditions.