 basketball leaves a player's hands at a height of 2.10 m
 above the floor. The basket is 3.05 m
 above the floor. The player likes to shoot the ball at a 40.0 ∘
 angle. If the shot is made from a horizontal distance of 12.00  m
  and must be accurate to  ±
 0.27  m
  (horizontally), what is the range of initial speeds allowed to make the basket?

This is a projectile motion problem. We want to calculate the range of initial speeds that would allow a basketball shot to be successful, given certain parameters. Let's break down the problem:

- Initial height of the ball: \( h_0 = 2.10 \, \text{m} \)
- Height of the basket: \( h_b = 3.05 \, \text{m} \)
- Horizontal distance to the basket: \( d = 12.00 \, \text{m} \)
- Shooting angle: \( \theta = 40^\circ \)
- Horizontal accuracy: \( \pm 0.27 \, \text{m} \), so the acceptable horizontal distance range is \( [11.73 \, \text{m}, 12.27 \, \text{m}] \)
- Gravitational acceleration: \( g = 9.81 \, \text{m/s}^2 \)

### Key Equations:

The projectile motion equations are:

1. **Horizontal motion:**
   \[
   x = v_0 \cos(\theta) \cdot t
   \]
   where \( v_0 \) is the initial velocity, and \( t \) is the time of flight.

2. **Vertical motion:**
   \[
   y = h_0 + v_0 \sin(\theta) \cdot t - \frac{1}{2} g t^2
   \]
   where \( y = h_b \) is the height of the basket.

We will first solve for the time of flight \( t \) and then express the initial velocity \( v_0 \) in terms of the known quantities. Finally, we will calculate the range of \( v_0 \) that satisfies the condition for horizontal accuracy.

### Step 1: Solving for Time of Flight

From the horizontal motion equation:
\[
t = \frac{x}{v_0 \cos(\theta)}
\]
Substitute this into the vertical motion equation to find \( v_0 \).

### Step 2: Calculating the Range of Initial Velocities

Now, I'll calculate the range of initial speeds using the bounds of horizontal accuracy.It seems I forgot to import the `tan` function. Let me correct that and recalculate the initial speeds.It seems like I can’t do more advanced data analysis right now. Please try again later. 

If you would like, I can explain how to solve the problem manually or help you with any other topic. Let me know how you'd like to proceed!

Here are five questions that expand on the topic of projectile motion:

1. How does air resistance affect projectile motion in real-life scenarios?
2. Can the angle of release influence the maximum height of a projectile?
3. What is the significance of the initial speed in determining the range of a projectile?
4. How does gravitational acceleration vary across different planets, and how does it affect projectile motion?
5. Can a projectile ever reach its target at the same height it was released if launched at an angle?

**Tip**: When solving projectile motion problems, always separate horizontal and vertical components and use them independently in calculations.

A basketball leaves a player's hands at a height of 2.10 m above the floor. The basket is 3.05 m above the floor. The player likes to shoot the ball at a 40.0° angle. If the shot is made from a horizontal distance of 12.00 m and must be accurate to ± 0.27 m (horizontally), what is the range of initial speeds allowed to make the basket?

Explore how to solve a projectile motion problem involving a basketball shot from 12m at a 40° angle. Learn the key concepts of kinematics and trigonometry, and how to calculate the range of initial speeds allowed to make the basket. Suitable for high school and college students.

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