**Maths prize 17 November 2017: ball and bucket**

Ashley drops a bucket from a height h above the ground. At the same moment Brenda, at ground level and at a horizontal distance s from the falling bucket, throws a ball with speed v at exactly the right angle θ to hit the bucket as it falls. Brenda’s height is negligible compared to h.

Find θ, the time t at which the ball hits the bucket, and the minimum v to be able to hit the bucket before it reaches the ground.

The Year 13 Further Maths class, collectively, with Tegan Hill doing much of the work, worked out how to find θ and t, and Mohaned al-Bassam at least started an attempt to find v. Jeffrey Sylvester made a good effort. But, disappointingly, no-one wrote up an attempted solution.

Take a simple special case: if *g* is negligibly small. Then:

The ball hits the bucket after it has dropped only a negligible distance.

Imagine *g* increases. The bucket falls, and the ball falls off vertically from the trajectory it would have with negligible *g*, both with the same acceleration *g*, and over the same time as each other. Therefore, as *g* increases from negligible amounts, the required angle θ remains the same, because the falling-down and the falling-off increase exactly in line with each other. Therefore, whatever *g*:

The time t= time taken by ball at horizontal speed v cos θ to travel distance s, so, given v, h, s , does not depend on *g*. So it is the same as it would be if *g* were negligible:

For the ball to hit the bucket before the ground,

, so:

For a much less neat way of doing this, see:

http://web.mit.edu/8.01t/www/materials/modules/chapter05.pdf

### Like this:

Like Loading...

*Related*