A canon shoots a ball upwards with an initial speed of 100 m/s. The total time of flight of the ball is 20 s before it hits the ground. The ball looses 70% of its speed after hitting the ground. Which among the following is the correct height that the ball will bounce up after its first bounce ? (g = 10 m/s2)

Correct — D, 45 m. The ball leaves the ground at 100 metres per second and the flight lasts 20 seconds, which is the symmetric figure for g = 10: it rises for 10 seconds, falls for 10 seconds and arrives back at the ground at 100 metres per second. Losing 70 per cent of its speed leaves 30 per cent, so it rebounds at 0.30 x 100 = 30 metres per second. For a body thrown straight up, maximum height h = v squared divided by 2g, which gives 30 x 30 divided by 20 = 900 divided by 20 = 45 metres. Note that the loss is stated as a loss of speed, not of energy — energy would fall to 9 per cent, and the height with it, and 45 metres is exactly 9 per cent of the original 500 metres.

• (a)100 m — 100 m is a mis-reading of the initial speed as a height. The launch speed is 100 metres per second, and the height it buys on the first ascent is 500 metres, not 100.
• (b)70 m — 70 m follows from keeping 70 per cent of the speed instead of losing it. The stem says the ball loses 70 per cent, so 30 per cent survives, and in any case 70 metres is not the height that even 70 metres per second would reach.
• (c)50 m — 50 m looks like a tenth of the original 500 metre rise, which is the answer you get by scaling the height by the surviving speed fraction instead of by its square. Height goes as the square of the speed, so the correct factor is 0.09, giving 45 metres.

Under constant gravity and with no air resistance a projectile's upward and downward journeys are mirror images: the time up equals the time down, and the landing speed equals the launch speed. Two standard results follow — total time of flight T = 2u divided by g, and maximum height h = u squared divided by 2g. A collision that scales the speed by a factor k scales the next height by k squared, because height depends on the square of the speed.

The two-line check on the data is worth doing first. With u = 100 and g = 10, T = 2 x 100 divided by 10 = 20 seconds, which is exactly the flight time printed, so the numbers are self-consistent and the return speed is certainly 100 metres per second. After that the only decision is what 'looses 70% of its speed' scales. It scales the speed, so multiply by 0.30, then square when you convert to height. As printed the paper spells the launcher 'canon' and the verb 'looses'; those are the booklet's own spellings and are reproduced here unchanged.

• Time of flight for a vertical throw: T = 2u divided by g. With u = 100 and g = 10 that is 20 seconds, matching the stem.
• Maximum height for a vertical throw: h = u squared divided by 2g. The first ascent here is 500 metres.
• A body launched upward returns to the same level with the same speed when air resistance is ignored.
• If a bounce keeps a fraction k of the speed, the next height is k squared times the previous one.
• Keeping 30 per cent of the speed keeps 9 per cent of the height: 0.09 x 500 = 45 metres.

• Scaling the height by 0.30 instead of by 0.30 squared.
• Reading 'loses 70 per cent' as 'retains 70 per cent'.
• Ignoring the printed flight time, which is the free consistency check on the data.

A two-stage numerical item: one stage of projectile kinematics, then one collision step where the fraction has to be squared to convert speed into height.

• practice — not a real PYQ

  A ball thrown vertically up with 20 m/s returns to the thrower. Taking g = 10 m/s squared, what is its total time of flight?

  • (a)2 s
  • (b)4 s
  • (c)6 s
  • (d)8 s

  Answer(b) 4 s — T = 2u/g = 40/10 = 4 seconds.

• practice — not a real PYQ

  A ball hits the floor at 10 m/s and rebounds at 5 m/s. What fraction of its kinetic energy has it kept?

  • (a)One half
  • (b)One quarter
  • (c)One eighth
  • (d)Three quarters

  Answer(b) One quarter — energy goes as the square of the speed, and one half squared is one quarter.