Math Problem Statement

Let's solve the problem step by step. The goal is to find the initial speed (v0v_0v0​) of the basketball when it leaves Noa's hand so that it will go through the hoop.

Given:

Height of the basketball hoop: h=3.05 mh = 3.05 , \text{m}h=3.05m

Release height of the ball: y0=2.0 my_0 = 2.0 , \text{m}y0​=2.0m

Horizontal distance to the hoop: x=6.6 mx = 6.6 , \text{m}x=6.6m

Angle of release: α=50∘\alpha = 50^\circα=50∘

Gravitational acceleration: g=10 m/s2g = 10 , \text{m/s}^2g=10m/s2

Solution

To solve for the initial speed v0v_0 of the basketball, we can break the problem down into two parts: horizontal and vertical motion. Since the motion is two-dimensional, we'll apply the principles of projectile motion.

Step 1: Horizontal Motion

The horizontal distance xx to the hoop is given by:

x=v0cos(α)tx = v_0 \cos(\alpha) \cdot t

Where:

  • v0v_0 is the initial speed of the ball.
  • α\alpha is the angle of release.
  • tt is the time of flight.

We can solve for tt in terms of v0v_0:

t=xv0cos(α)t = \frac{x}{v_0 \cos(\alpha)}

Step 2: Vertical Motion

The vertical motion equation can be written as:

y=y0+v0sin(α)t12gt2y = y_0 + v_0 \sin(\alpha) \cdot t - \frac{1}{2} g t^2

Where:

  • yy is the height of the basketball hoop.
  • y0y_0 is the release height.
  • gg is the gravitational acceleration.

Given that y=hy = h, we can substitute and solve for tt using the equation for tt derived from the horizontal motion:

h=y0+v0sin(α)t12gt2h = y_0 + v_0 \sin(\alpha) \cdot t - \frac{1}{2} g t^2

Substituting t=xv0cos(α)t = \frac{x}{v_0 \cos(\alpha)}:

h=y0+v0sin(α)xv0cos(α)12g(xv0cos(α))2h = y_0 + v_0 \sin(\alpha) \cdot \frac{x}{v_0 \cos(\alpha)} - \frac{1}{2} g \left(\frac{x}{v_0 \cos(\alpha)}\right)^2

Simplifying, we get:

h=y0+xtan(α)gx22v02cos2(α)h = y_0 + x \tan(\alpha) - \frac{g x^2}{2 v_0^2 \cos^2(\alpha)}

Step 3: Solve for v0v_0

Now, solve for v0v_0:

hy0=xtan(α)gx22v02cos2(α)h - y_0 = x \tan(\alpha) - \frac{g x^2}{2 v_0^2 \cos^2(\alpha)}

Rearrange to isolate v02v_0^2:

gx22v02cos2(α)=xtan(α)(hy0)\frac{g x^2}{2 v_0^2 \cos^2(\alpha)} = x \tan(\alpha) - (h - y_0)

v02=gx22cos2(α)(xtan(α)(hy0))v_0^2 = \frac{g x^2}{2 \cos^2(\alpha) \left( x \tan(\alpha) - (h - y_0) \right)}

Finally, solve for v0v_0:

v0=gx22cos2(α)(xtan(α)(hy0))v_0 = \sqrt{\frac{g x^2}{2 \cos^2(\alpha) \left( x \tan(\alpha) - (h - y_0) \right)}}

Step 4: Substitute the Given Values

Given:

  • h=3.05mh = 3.05 \, \text{m}
  • y0=2.0my_0 = 2.0 \, \text{m}
  • x=6.6mx = 6.6 \, \text{m}
  • α=50\alpha = 50^\circ
  • g=10m/s2g = 10 \, \text{m/s}^2

Substitute these values:

v0=10×(6.6)22×cos2(50)×(6.6×tan(50)(3.052.0))v_0 = \sqrt{\frac{10 \times (6.6)^2}{2 \times \cos^2(50^\circ) \times \left( 6.6 \times \tan(50^\circ) - (3.05 - 2.0) \right)}}

Now, let's calculate the value.The initial speed v0v_0 of the basketball when it leaves Noa's hand should be approximately 8.79 m/s for the ball to go through the hoop.

Would you like any further details or explanations on this problem?

Here are five related questions you might find interesting:

  1. How does the angle of release affect the required initial speed?
  2. What would happen to the initial speed if the horizontal distance to the hoop increased?
  3. How would air resistance alter the required initial speed in this scenario?
  4. Can this calculation be adjusted for different gravitational conditions, like on the Moon?
  5. How would the initial speed change if the release height was higher?

Tip: Remember that increasing the angle of release will increase the time the ball stays in the air, but it also requires a higher initial speed to cover the same horizontal distance.

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Math Problem Analysis

Mathematical Concepts

Projectile Motion
Trigonometry

Formulas

Projectile motion equations
Trigonometric functions

Theorems

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Suitable Grade Level

Grades 10-12