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 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 to the hoop is given by:
Where:
- is the initial speed of the ball.
- is the angle of release.
- is the time of flight.
We can solve for in terms of :
Step 2: Vertical Motion
The vertical motion equation can be written as:
Where:
- is the height of the basketball hoop.
- is the release height.
- is the gravitational acceleration.
Given that , we can substitute and solve for using the equation for derived from the horizontal motion:
Substituting :
Simplifying, we get:
Step 3: Solve for
Now, solve for :
Rearrange to isolate :
Finally, solve for :
Step 4: Substitute the Given Values
Given:
Substitute these values:
Now, let's calculate the value.The initial speed 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:
- How does the angle of release affect the required initial speed?
- What would happen to the initial speed if the horizontal distance to the hoop increased?
- How would air resistance alter the required initial speed in this scenario?
- Can this calculation be adjusted for different gravitational conditions, like on the Moon?
- 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
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