A circus performer is juggling balls such that by the time a ball reaches maximum height next ball leaves the hand of juggler. He throws each ball vertically upward with the same initial velocity. If he throws n balls per second, and the time between throwing each ball is the same, what is the initial velocity of each ball?

(Assume little movement of juggler's hand)

To solve this problem, let's break it down step by step:

### Step 1: Time Between Throwing Each Ball
The juggler throws \( n \) balls per second. This means the time interval between each throw is:

\[
\text{Time between throws} = \frac{1}{n} \, \text{seconds}
\]

### Step 2: Total Time of Flight for a Ball
Since the next ball is thrown when the previous ball reaches its maximum height, the total time a ball spends in the air is twice the time it takes to reach its maximum height. Let \( T \) be the total time a ball is in the air. Then:

\[
T = 2 \times t_{\text{up}}
\]
where \( t_{\text{up}} \) is the time taken for the ball to reach the maximum height.

Also, we know that by the time the ball reaches its maximum height, another ball is thrown. Hence, the total time of flight must be equal to the time it takes to throw \( n \) balls. Therefore:

\[
T = \frac{1}{n}
\]

### Step 3: Time to Reach Maximum Height
We know the time to reach maximum height for a vertically thrown object is given by the equation:

\[
t_{\text{up}} = \frac{v_0}{g}
\]
where:
- \( v_0 \) is the initial velocity of the ball,
- \( g \) is the acceleration due to gravity (\( g = 9.8 \, \text{m/s}^2 \)).

Since \( T = 2 \times t_{\text{up}} \), we can substitute \( T \) into this equation:

\[
T = 2 \times \frac{v_0}{g}
\]
And from Step 2, we know \( T = \frac{1}{n} \). Thus:

\[
\frac{1}{n} = 2 \times \frac{v_0}{g}
\]

### Step 4: Solve for \( v_0 \)
Now, solve for \( v_0 \):

\[
v_0 = \frac{g}{2n}
\]

### Final Answer
The initial velocity of each ball is:

\[
v_0 = \frac{9.8}{2n} \, \text{m/s}
\]

### Tip
The number of balls thrown per second, \( n \), directly impacts the required velocity: the faster the juggler throws balls, the lower the initial velocity needed for each ball to ensure it reaches its peak before the next ball is thrown.

Let me know if you'd like more details or further clarification!

### Further Questions:
1. How would the equation change if the acceleration due to gravity was different?
2. What happens to the velocity if the juggler increases the number of balls thrown per second?
3. If the juggler reduces the number of balls thrown to half per second, how does the initial velocity change?
4. How does the height reached by each ball relate to the initial velocity?
5. How would air resistance affect the initial velocity calculation?

A circus performer is juggling balls such that by the time a ball reaches maximum height next ball leaves the hand of juggler. He throws each ball vertically upward with the same initial velocity. If he throws n balls per second, and the time between throwing each ball is the same, what is the initial velocity of each ball?

This problem demonstrates how to calculate the initial velocity of a vertically thrown ball by a juggler using projectile motion concepts. Learn how the number of balls thrown per second (n) influences the velocity needed for each ball to reach its peak before the next throw. Suitable for high school students in Grades 10-12.

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