A First Course in Physics · Ch. II, §§ 38–48 · Chapter 2: Force and Motion
5. Falling bodies and accelerated motion
Goal: Your student can define velocity and acceleration, use v = at and S = ½at² for falling bodies, and explain Galileo's ramp and tower experiments.
You'll need
- The book, open to these pages
- A long board or plastic rain gutter for a ramp, propped up a few inches at one end
- A marble or small ball
- A stopwatch or phone timer; a metronome app if you have one
- Tape measure and masking tape
- A coin and a small piece of paper
- A notebook and pencil
- A calculator (optional, for the problems)
What's in the book
Uniform motion, velocity and acceleration; Galileo's grooved-board experiment timed with a metronome; the table of falling bodies (distances 1, 4, 9, 16 …); the formulas v = at, s = ½a(2t − 1), S = ½at²; the acceleration of gravity (980 cm per sec per sec); the leaning tower of Pisa and the feather-and-coin tube; and the height reached by a body thrown upward. Eleven problems follow.
The lesson, step by step
Hook question
SayIf a ball rolls 10 cm in the first second down a ramp, how far will it roll in 2 seconds? Most people guess 20 cm. Galileo found something surprising.
Read together
Read §§ 38–48. Galileo's discovery:
He found that a body falls exactly 4 times as far in 2 seconds as in 1, 9 times as far in 3 seconds, 16 times as far in 4 seconds
Try it: Galileo's ramp
Prop the board or gutter so it rises only a few inches over its whole length. Let the marble go at a beat of the metronome (or say "tick" every second). Put tape where it is after 1 second, then after 2 and 3 seconds. Measure from the start.
The distances should be roughly 1 : 4 : 9. It takes a few tries; slow ramps work best.
Explain the formulas
Acceleration is the velocity gained each second. For anything falling freely, a = 980 cm per second each second (about 32 ft). In SI units that is 9.8 m/s².
v = at (speed after t seconds); S = ½at² (distance in t seconds); and the book shows v = √(2gS) for a body falling from height S. Work the example in § 44 together: a ball that rolls 10 cm in the first second has a = 20, and after 10 seconds has gone ½ × 20 × 100 = 1000 cm.
Try it: feather and coin
Drop a coin and a flat sheet of paper together: the coin wins. Now crumple the paper into a tight ball and try again. Then lay a small flat piece of paper on top of the coin and drop them together. The paper stays with the coin, because the coin keeps the air off it. This shows what the vacuum tube in § 46 shows.
Say it back
Your student explains what Galileo proved at Pisa and why a feather falls slowly in air.
Problems
Work problems 1–11 on book pages 31–32. Use g = 980 cm/sec² (or 32.16 ft/sec² when the problem is in feet).
Afterwards
Drop a ball from a table and from twice the height. Does it take twice as long? (No: about 1.4 times as long.)
Tip
Historians debate whether Galileo really dropped balls from the tower at Pisa; the ramp experiments are well recorded.
Answers to the book's problems
1. About 11.4 sec. 2. 176 sec (2 min 56 sec). 3. 49,000 cm = 490 m. 4. 7840 cm/sec. 5. 4410 cm (½ × 980 × 9). 6. 122.5 cm. 7. 14,000 cm/sec = 140 m/sec. 8. About 459,000 cm, roughly 4590 m. 9. About 189 ft./sec (√(2 × 32.16 × 555)). 10. About 167 cm/sec. 11. The acceleration down the plane is 80 cm/sec², and it equals g × 40.8/500, so g ≈ 980 cm/sec².
Worked out for this site from the scan. If a number in your copy differs, trust the book.