When the juggler throws the first ball in air it will take 4 sec to get to the height of 80 metres (by s=ut+1/2at square) after some times we assume that any one ball is at the top i.e at 80 metres and the juggler has one ball in his hand that means the rest of the 8 balls are in air we consider that 4 balls are going to the top and 4 are coming to the jugglers hand now we know total time to reach the top or from the top to jugglers hand is 4 sec and hence for the 4 balls to not collide and move smoothly there should be and equal interval between each of them that is 0.8 s.............. Hope it helps

Maintaining Distance between Juggled Balls

To maintain a proper distance between the 10 balls in motion, the juggler needs to carefully time the release of each ball. This is crucial to ensure that they do not collide or come too close to each other during their trajectory.

Physics of Juggling

Juggling involves the principles of projectile motion. When the juggler throws a ball into the air, it follows a parabolic path due to the combined effect of its initial velocity and the force of gravity. The time of flight and the height reached by the ball depend on the initial velocity and the angle of projection.

Calculating Time Interval

To maintain a distance between the balls, the juggler needs to ensure that each ball is released at the right time. This can be achieved by calculating the time interval between successive releases. Here's how it can be done:

1. Determine the height reached by the ball: In this case, each ball rises to a height of 80 meters from the juggler's hands.

2. Calculate the time taken to reach that height: Using the principles of projectile motion, the time taken for a ball to reach a certain height can be calculated. Let's assume the initial velocity of the ball is v m/s. The time taken to reach a height h meters can be calculated using the equation h = (1/2)gt^2, where g is the acceleration due to gravity (approximately 9.8 m/s^2) and t is the time taken. Rearranging the equation, we get t = sqrt(2h/g).

3. Determine the time interval: The time interval between releases should be equal to or greater than the time taken for a ball to reach its maximum height. This ensures that the balls do not collide or come too close to each other. Therefore, the time interval should be equal to or greater than the calculated time taken.

Example Calculation

Let's assume the initial velocity of the balls is 10 m/s. Using the equation t = sqrt(2h/g), the time taken for a ball to reach a height of 80 meters can be calculated as follows:

t = sqrt(2 * 80 / 9.8) = sqrt(160 / 9.8) ≈ 4.04 seconds.

Therefore, the time interval between the releases of the balls should be equal to or greater than 4.04 seconds.

Conclusion

To maintain a proper distance between the 10 balls in motion, the juggler needs to carefully time their releases. By calculating the time taken for a ball to reach its maximum height and ensuring that the time interval between releases is equal to or greater than this value, the juggler can keep the balls from colliding or coming too close to each other. This allows for a visually appealing and skillful juggling performance.