Two particles: gravity, a track, successive intervals and mid-air collisions

Last time

Before we start: last time, in Variable acceleration in stages and velocity as a function of time, you learned these points. Here they are again on one card. Keep them in mind, because today builds on them.

Two particles, one equation

Two particles, one story. Each has its own displacement formula. The question links them with one fact: they meet, they land together, or one starts later. Write both formulas with the same clock, set the linking fact, and you have an equation to solve.

The method

First choose one time variable and one positive direction. Write each particle's displacement using suvat. Second, translate the link into an equation: equal displacements, displacements adding to a gap, or equal times. Third, solve, then check that every time is positive and the particle really is moving at that time.

A later start

Two points A and B are at the same height h. P falls from rest at A. One second later, Q is thrown down from B at 18 metres per second. They land together. P falls for t plus 1 seconds: h equals 5 times t plus 1, all squared. Q falls for t: h equals 18 t plus 5 t squared. Expand: 5 t squared plus 10 t plus 5 equals 18 t plus 5 t squared. The t squared terms cancel, so 8 t equals 5 and t is 5 over 8. Then h is 18 times 5 over 8, plus 5 times 25 over 64, which is 13.2 metres.