From the top of a cliff overlooking a lake, a person throws two stones, as shown in the drawing. The cliff is 39.2 m high. The two stones described have identical initial speeds of v0 = 19.2 m/s and are thrown at an angle θ = 29.2 °, one below the horizontal and one above the horizontal. What is the distance between the points where the stones strike the water? Neglect air resistance.

Question

From the top of a cliff overlooking a lake, a person throws two stones, as shown in the drawing. The cliff is 39.2 m high. The two stones described have identical initial speeds of v0 = 19.2 m/s and are thrown at an angle θ = 29.2 °, one below the horizontal and one above the horizontal. What is the distance between the points where the stones strike the water? Neglect air resistance.

Answer 1

To find the distance between the points where the stones strike the water, we can break down the motion of each stone into horizontal and vertical components.

1. For the stone thrown below the horizontal:
- The initial vertical velocity (vy) can be calculated using the formula vy = v0 * sin(θ).
- The initial horizontal velocity (vx) can be calculated using the formula vx = v0 * cos(θ).
- The time taken (t) for the stone to hit the water can be calculated using the formula t = (2 * vy) / g, where g is the acceleration due to gravity (approximately 9.8 m/s^2).
- The horizontal distance (dx) traveled by the stone can be calculated using the formula dx = vx * t.

2. For the stone thrown above the horizontal:
- The initial vertical velocity (vy) can be calculated using the formula vy = v0 * sin(θ).
- The initial horizontal velocity (vx) can be calculated using the formula vx = v0 * cos(θ).
- The time taken (t) for the stone to hit the water can be calculated using the formula t = (2 * vy) / g, where g is the acceleration due to gravity (approximately 9.8 m/s^2).
- The horizontal distance (dx) traveled by the stone can be calculated using the formula dx = vx * t.

3. The total distance between the points where the stones strike the water is the sum of the horizontal distances traveled by each stone.

Let's calculate the distances.

Calculations for the stone thrown below the horizontal:
vy = v0 * sin(θ) = 19.2 m/s * sin(29.2°) ≈ 9.752 m/s
vx = v0 * cos(θ) = 19.2 m/s * cos(29.2°) ≈ 16.934 m/s
t = (2 * vy) / g = (2 * 9.752 m/s) / 9.8 m/s^2 ≈ 1.976 s
dx1 = vx * t = 16.934 m/s * 1.976 s ≈ 33.493 m

Calculations for the stone thrown above the horizontal:
vy = v0 * sin(θ) = 19.2 m/s * sin(29.2°) ≈ 9.752 m/s
vx = v0 * cos(θ) = 19.2 m/s * cos(29.2°) ≈ 16.934 m/s
t = (2 * vy) / g = (2 * 9.752 m/s) / 9.8 m/s^2 ≈ 1.976 s
dx2 = vx * t = 16.934 m/s * 1.976 s ≈ 33.493 m

Total distance between the points where the stones strike the water:
distance = dx1 + dx2 = 33.493 m + 33.493 m = 66.986 m

Therefore, the distance between the points where the stones strike the water is approximately 66.986 meters.

Answer 2

To solve this problem, we need to find the horizontal distances traveled by the two stones and calculate the difference between them to determine the distance between the points where the stones strike the water.

Here's how you can proceed:

1. First, let's find the time of flight for each stone. The time of flight is the total time taken by a stone to reach the water from the top of the cliff.

The formula for the time of flight is:
time of flight = (2 * initial vertical velocity) / gravitational acceleration

Since the stones are thrown at the same speed and angle, they will have the same initial vertical velocity. So, we can find the time of flight using the formula above.

initial vertical velocity = initial speed * sin(angle)

For the given values:
initial speed = 19.2 m/s
angle = 29.2 degrees

First, convert the angle to radians:
angle_radians = angle * pi / 180

Then calculate the initial vertical velocity:
initial vertical velocity = initial speed * sin(angle_radians)

2. Once we have the time of flight, we can calculate the horizontal distances traveled by the stones.

The formula for the horizontal distance is:
horizontal distance = initial speed * cos(angle) * time of flight

For each stone, use the respective angle and time of flight to calculate the horizontal distance.

3. Finally, subtract the two horizontal distances to find the distance between the points where the stones strike the water.

Let's calculate these values step by step:

Step 1: Calculate the initial vertical velocity:
angle_radians = 29.2 * pi / 180
initial vertical velocity = 19.2 * sin(angle_radians)

Step 2: Calculate the time of flight:
time of flight = (2 * initial vertical velocity) / gravitational acceleration

Step 3: Calculate the horizontal distance for each stone:
horizontal distance for stone below = 19.2 * cos(angle) * time of flight
horizontal distance for stone above = 19.2 * cos(angle) * time of flight

Step 4: Calculate the difference between the two horizontal distances:
distance between the points = abs(horizontal distance for stone below - horizontal distance for stone above)

Now, you can substitute the values into these formulas and calculate the result.

Answer 3

The two will have the same x velocities (19.2 cos29.2), and the distance traveled will be that number times t. Find the two times by solving in y:

.5*-9.8t^2 + 19.2sin29.2t = -39.2
and
.5*-9.8t^2 - 19.2sin29.2 = -39.2