2) A ball is hit at an angle of 36 degrees to the horizontal from 14 feet above the ground. If it hits the ground 26 feet away (in horizontal distance), what was the initial velocity? Assume there is no air resistance.
To find the initial velocity of the ball, you can use the equations of motion. The key here is to recognize that the horizontal and vertical components of motion are independent of each other.
Let's break down this problem step by step:
1. Start by identifying the known values:
- The angle of projection, θ = 36 degrees
- The vertical distance, h = 14 feet
- The horizontal distance, d = 26 feet
- Acceleration due to gravity, g = 32.2 ft/s² (assuming you are working in the US customary system)
2. First, calculate the time of flight:
- The formula for the time of flight is given by:
t = 2 * (V₀ * sin(θ)) / g
where V₀ is the initial velocity.
- Since the ball is projected at an angle and then lands on the ground, the time of flight for the ball's vertical motion will be twice the time it takes to reach the maximum height.
- The time to reach the maximum height is given by:
t₁ = V₀ * sin(θ) / g
- The total time of flight is then:
t = 2 * t₁
3. Next, calculate the horizontal component of the initial velocity:
- The horizontal component of motion remains constant throughout the entire trajectory.
- The horizontal distance traveled by the ball is given by:
d = V₀ * cos(θ) * t
- Rearranging this equation, we can solve for V₀:
V₀ = d / (cos(θ) * t)
4. Finally, substitute the values into the equation to find the initial velocity:
- Plug in the given values:
θ = 36 degrees, h = 14 feet, d = 26 feet, g = 32.2 ft/s²
- Use the equations from steps 2 and 3 to calculate V₀.
Following these steps, you can find the initial velocity of the ball.