Samantha has the opportunity to win $1 million by sinking a half court shot. She shoots the basketball with a speed of 53.1 ft/sft/s while standing 47 ftft away from the basketball hoop. The hoop is 10 ftft above the floor and the basketball leaves her hand 7.3 ftft above the floor. Determine the smaller of the two angles that Samantha can shoot the basketball to win the prize.

To solve this problem, we can model the situation using the physics of projectile motion. We need to find the smaller of two possible launch angles that will allow Samantha to make the basketball into the hoop.

### Step-by-Step Approach:

1. **Known Variables:**
   - Initial velocity $ v_0 = 53.1 \, \text{ft/s} $
   - Horizontal distance to the hoop $ x = 47 \, \text{ft} $
   - Height difference between release point and hoop: $ \Delta y = 10 - 7.3 = 2.7 \, \text{ft} $
   - Gravitational acceleration $ g = 32.2 \, \text{ft/s}^2 $

2. **Projectile Motion Equations:**

The horizontal and vertical components of the motion are described by the following kinematic equations:

\[
x = v_0 \cos(\theta) t
\]

\[
y = y_0 + v_0 \sin(\theta) t - \frac{1}{2} g t^2
\]

Where:
- $ x $ is the horizontal distance to the hoop.
- $ y_0 = 7.3 \, \text{ft} $ is the initial height.
- $ \theta $ is the angle of the shot (what we're trying to find).
- $ t $ is the time of flight.

We want the ball to reach the basket when $ y = 10 \, \text{ft} $ and $ x = 47 \, \text{ft} $.

### Step 1: Solving for time of flight $ t $

From the horizontal motion equation:

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

Substitute $ x = 47 \, \text{ft} $ and $ v_0 = 53.1 \, \text{ft/s} $:

\[
t = \frac{47}{53.1 \cos(\theta)}
\]

### Step 2: Substituting into the vertical motion equation

Next, we substitute $ t $ into the vertical motion equation:

\[
10 = 7.3 + 53.1 \sin(\theta) \left( \frac{47}{53.1 \cos(\theta)} \right) - \frac{1}{2} (32.2) \left( \frac{47}{53.1 \cos(\theta)} \right)^2
\]

Simplify this equation step by step to isolate $ \theta $, and solve for the smaller of the two possible angles.

Let me calculate the value of the smaller angle.The smaller angle that Samantha can use to shoot the basketball and win the prize is approximately $ 19.84^\circ $.

Would you like more details or have any questions about the calculation?

Here are 5 related questions to expand on this topic:
1. How would the angle change if the distance to the hoop was increased?
2. What would happen if Samantha shot the ball with a different initial speed?
3. How would air resistance affect the outcome of this shot?
4. What is the larger angle that Samantha can use to make the shot?
5. How can we calculate the time of flight for this shot?

**Tip:** In projectile motion problems, the same initial speed can give two different angles that lead to the same target because one trajectory is higher (but longer in time) and the other is lower (shorter in time).

Samantha has the opportunity to win $1 million by sinking a half court shot. She shoots the basketball with a speed of 53.1 ft/s while standing 47 ft away from the basketball hoop. The hoop is 10 ft above the floor and the basketball leaves her hand 7.3 ft above the floor. Determine the smaller of the two angles that Samantha can shoot the basketball to win the prize.

Learn how to determine the smaller of two angles for a half court basketball shot using projectile motion. Samantha's shot involves a speed of 53.1 ft/s, a distance of 47 ft to the hoop, and a release height of 7.3 ft. This detailed guide covers kinematic equations, trigonometry, and solving for the time of flight and angle. Suitable for Grades 10-12 students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation