To find the angle at which the ball was hit, we need to solve the problem using the equations of motion under constant acceleration. Here's the breakdown:

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Given:

• Initial speed ($v_0$) = $30 \, \text{m/s}$
• Horizontal distance to the fence ($x$) = $80 \, \text{m}$
• Height of the fence ($y$) = $1.5 \, \text{m}$
• Gravitational acceleration ($g$) = $9.8 \, \text{m/s}^2$

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Equations of Motion:

1. Horizontal motion: $x = v_0 \cos(\theta) \cdot t$ where $t$ is the time of flight.

2. Vertical motion: $y = v_0 \sin(\theta) \cdot t - \frac{1}{2} g t^2$

We solve these equations simultaneously to find $\theta$.

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Step 1: Solve for $t$ from horizontal motion

$t = \frac{x}{v_0 \cos(\theta)}$

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Step 2: Substitute $t$ into vertical motion equation

$y = v_0 \sin(\theta) \cdot \frac{x}{v_0 \cos(\theta)} - \frac{1}{2} g \left(\frac{x}{v_0 \cos(\theta)}\right)^2$ Simplify: $y = x \tan(\theta) - \frac{g x^2}{2 v_0^2 \cos^2(\theta)}$

Substitute $y = 1.5 \, \text{m}$, $x = 80 \, \text{m}$, $v_0 = 30 \, \text{m/s}$, and $g = 9.8 \, \text{m/s}^2$: $1.5 = 80 \tan(\theta) - \frac{9.8 \cdot 80^2}{2 \cdot 30^2 \cos^2(\theta)}$

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Step 3: Solve for $\theta$

This is a nonlinear equation that requires numerical or iterative methods. Let’s solve it. from scipy.optimize import fsolve

Given constants

v0 = 30 # initial speed (m/s) x = 80 # horizontal distance (m) y = 1.5 # height of the fence (m) g = 9.8 # acceleration due to gravity (m/s^2)

Equation to solve

def equation(theta): theta_rad = np.radians(theta) # Convert angle to radians term1 = x * np.tan(theta_rad) term2 = (g * x2) / (2 * v02 * np.cos(theta_rad)**2) return term1 - term2 - y

Initial guess for the angle

initial_guess = 45 # degrees

Solve for theta

theta_solution = fsolve(equation, initial_guess)[0] The angle at which the ball was hit is approximately 31.96°.

Would you like a detailed step-by-step explanation of the solution or have additional questions?

Relative Questions:

1. How does air resistance affect the trajectory of the ball in real-life scenarios?
2. Can this method be applied to other projectile problems with different initial conditions?
3. What assumptions are made in deriving the equations of motion for this problem?
4. How would the angle change if the fence were closer or farther away?
5. How does the initial speed of the ball influence the required launch angle?

Tip:

Always check units and assumptions in physics