Before a systematic study of the behavior of matter can begin, we must understand the nature of motion. The most fundamental relationship is simply the definition of velocity:

		distance (S)
velocity (V)	=	------------
		time (T)

The uninitiated may wonder why ";s"; is used for distance. Wonder no more. It is, because it is. To ease the communications problem, certain letters (sometimes Greek ones) are typically used to designate particular physical quantities. An example we've already seen is the use of q to designate an angle, the use of r to designate density, and now the use of s to designate distance. Don't fight it, that would be a losing battle.

Now I must digress for a moment to keep the purists off my back. Velocity is not the same thing as speed. 55 mi/hr is a speed, while 55 mi/hr to the north is a velocity. Speed is a scalar quantity and has only magnitude, or size, associated with it. Velocity is a vector quantity and has both magnitude and direction. Thus a vector requires two numbers to specify it while a scalar requires but one. (e.g.: 30 m/sec at 55° as opposed to just 30 m/sec)

A word to the wise: the difference between a scalar and a vector is an important one in physics, not generally appreciated by the public. To prevent appearing gauche, don't jump to correct others who use speed and velocity interchangeably. Even physicists get sloppy at times with these words. The equation v = s/t actually yields speed, and should be written v = s/t to properly signify the vector nature of velocity.

The easy way to remember this equation is s = vt. Use it below.

1. A car travels at 6.3 m/sec for 8.5 seconds. How far does it go?

s = vt s = (6.3 m/sec) (8.5 sec) s = 53.6 m

2. A jet travels at 480 m/sec.
a) How long does it take to travel 200 m?
b) How long to travel 9.4 km?
c) How long to travel 6 inches?

a) t = s/v t = 200 m / 480 m/sec t = .416 sec

b) t = s/v t = [(9.4 km) (103 m / 1 km)] / 480 m/sec t = 19.6 sec

c) t = s/v t = [(6 in) (2.54 cm / 1 in)] / 480 m/sec t = 3.18 * 10-4 sec

3. A cockroach completes one lap of a plate in 12.3 seconds. If the radius of the plate is 16.1 cm, how fast was he traveling?

s = vt v = s/t v = [2(3.14)r]/t v = 2(3.14) (16.1 cm) / 12.3 sec v = 8.22 cm/sec

4. Roger Overndout rides a ferris wheel with a 12 m radius, revolving at 0.22 rad/sec.
a) How far will Roger travel in 1 second?
b) What is his linear speed?

a) In 1 seconds the wheel revolves 0.22 rad thus he travels: s = rØ s = (12 m) (.22 rad) s = 2.64 m

b) Roger travels 2.64 m in 1 sec: so v = s/t v = 2.64 m / 1 sec v = 2.64 m/sec [easier: v = wr = (.22 rad/sec) (12 m) = 2.64 m/sec]

5. Jeanette Ikode has stuck her '57 Ford in the mud. As she spins her wheels, a passer-by notes that the treads of her tires are kicking mud out at 18 ft/sec. The radius of her tires is 15 inches. At what angular velocity is the wheel turning? (i.e.: Through how many radians does it turn in one second?)

(18 ft/sec)(12 in / 1 ft) = 216 in/sec thus, in one second, the wheel turns 216 in Ø = s/r Ø = 216/15 Ø = 14.4 rad so the angular velocity is: w = 14.4 rad/sec

6. A caterpillar and a snail meet at the 37° angle of the triangle shown. They decide to race to opposite vertices, the snail along the hypotenuse while the caterpillar runs along the baseline. If the caterpillar runs at 1.7 cm/sec, and the snail at 2.0 cm/sec, which will arrive first?

tcaterpillar = s/v tcaterpillar = 22 cm / 1.7 cm/sec tcaterpillar = 12.9 sec tsnail = s/v tsnail = (22 / cos37) / (2.0 cm/sec) tsnail = 13.8 sec so the caterpillar wins.

7. To check the balance on his bike wheel, Justin Case gives it a spin. If its radius is 13.5 inch, and it spins at 5.8 rad/sec, what is the linear velocity of a point on the rim?

In one second the wheel rotates 5.8 rad, so a point on the rim travels: s = rØ s = (13.5 in) (5.8) s = 78.3 in

8. Two ants race across the top of a 28.1 cm long Wheaties box. One travels at 4.2 cm/sec while the other drags along at 3.9 cm/sec. When the first one crosses the finish line, how far behind is the second one?

First we must find out how much time has elapsed. t = s/v t = 28.1 cm / 4.2 cm/sec t = 6.69 sec Now we find how far the second ant got in this time. s = vt s = (3.9 cm/sec) (6.69 sec) s = 26.1 cm or 2.0 cm back

9. A blade of grass is found to have grown 2.37 mm in a 24 hour period. What is its average rate of growth in meters/second?

s = vt v = s/t v = (2.37 mm / 24 hr) (1 m / 103 mm) (1 hr / 60 min) (1 min / 60 sec) v = 2.74 * 10-8 m/sec

10. Augusta Wind has a small boat that, in still water, travels at 3.5 m/sec. He boats to town along a river which flows at 1.6 m/sec. How long will it take him if the town is 3.6 km upstream from his home? .....How long will it take him to get back?

a) Upstream, his speed is: 3.5 m/s - 1.6 m/s = 1.9 m/sec But t = s/v t = 3.6 * 103 m / (1.9 m/s) t = 1.89 * 103 sec

b) Downstream: v = 3.5 + 1.6 v = 5.1 m/s t = s/v t = 3.6 * 103 / (5.1 m/s) t = 706 sec

11*. Agatha Ant starts at one edge of a place mat, 29.6 cm wide. She travels at 1.4 cm/sec at 52° from one edge. How long will it take her to get across?

The trick here is to recognized that while agatha's velocity is 1.4 cm/sec, she is advancing toward the opposite side at less than this speed. Here is her true speed relative to the placemat here is the rate at which she is approaching the opposite side. (this is the one we're interested in) From trig: sinØ = opp/hyp sin52 = vy / (1.4 cm/s) vy = (1.4 cm/s) (sin52) vy = 1.10 cm/s But t = s / vy t = 29.6 cm / (1.10 cm/s) t = 26.8 sec

12. A speeding bumble bee flies 34° west of north at 0.42 m/sec. What are the westerly and northerly components of his motion?

First sketch the components: cosØ = vy / v sinØ = vx / v cos34° = vy / (.42 m/s) vx = .42sin34 vy = .42cos34 vy = .348 m/s vx = .235 m/s

13. A carpenter uses a hand saw to cut through a piece of plywood, 4 ft wide. He cuts at a 41° angle to the edge and cuts at a steady rate of 0.84 cm/sec. How long will it take him to cut through?

The component of motion we want is vy sin41 = opp/hyp sin41 = vy/v sin41 = vy/(.84 cm/sec) So vy = (.84 cm/sec) (sin41) vy = .55 cm/sec But t = sy/vy t = [(4 ft) (12 in / 1 ft) (2.54 cm / 1 in)] / (.55 cm/sec) t = 222 sec

14*. Noah Formula takes off from a pier on a lake in his newly completed boat. If the pier is 7.0 m long and Noah travels at 2.8 m/sec at 34° as shown, how far from shore will he be in 16.3 seconds?

We must first find the component of velocity away from shore: sin34 = vy / (2.8 m/s) vy = (2.8 m/s) (sin34) vy = 1.57 m/s sy = vyt + so sy = (1.57 m/sec) (16.3 sec) + 7.0 m sy = 32.6 m

15**. General Lee Friendly is to fly an airplane into enemy territory, 115 miles away. His plane travels at 85 mi/hr, and is to cross over the enemy line in precisely 1.8 hours. At what angle should he aim his plane for this to work out?

First find the somponent of his motion in the direction of the enemy: cosØ = vy/85 vy= 85cosØ But sy = vyt 115 = (85cosØ) (1.8 sec) Hence cosØ = 115 / (85) (1.8) cosØ = .75 Ø = 41.3°

16. A slug, his mind impaired by eating drug-laced cookies, wanders in a circle of 8.6 cm radius. If he makes 2.5 laps in 75 seconds, what is his linear speed? (";linear"; means how fast in centimeters/second as opposed to radians/second?.)

s = vt v = s/t v = (2.5 laps) [2(3.14) * 8.6 cm/lap] / 75 sec v =1.80 cm/sec

17. A rock on a string is swung in a circle of 1.3 m radius. If it travels at 2.4 m/sec, how long will it need to complete exactly 8 orbits?

s = vt t = s/v t = [[2(3.14) * 1.3 m] (8)] / (2.4 m/sec) t = 27.2 sec

18. The moon completes one orbit of Earth in 27.3 days. If its orbit has a radius of 3.84 x l05 km, how far does it travel in 3.0 seconds?

First we need the velocity of the moon: v = s/t v = [2(3.14)r]/t v = [2(3.14)(3.84 * 105 km)] / [(27.3 d) (24 hr/1 d) (60 min/1 hr) (60 sec/1 min)] v = 1.02 km/sec Here we use the distance [2(3.14)r] and the time of one orbit. But s = vt s = (1.02 km/sec) (3.0 sec) s = 3.06 km

19*. Two bees have a race to the hive. Bee A starts 40.0 m ahead of bee B. If A travels at 13.6 m/sec while B travels at 15.0 m/sec,
a) how long will it take for B to pass A?
b) how far will B have traveled?

a) This is a problem with 2 unknowns (s and t) so it will require 2 equations. First write an expession for the position of A: sA = vt + so sA = 13.6t + 40 Think: at t = 0, A is at 40. At t = 2 sec, A is at 67.2 likewise: sB = vt + so sB = 15.0t + 0 The bees pass when sA = sB: 13.6t + 40 = 15t + 0 t = 28.6 sec

b) At t = 28.6 sec: sB = 15t sB = 15 * 28.6 sB = 429 m

20*. Two electric trains are on a collision course. Engine A travels to the right at 19.5 cm/sec. Engine B, 1.8 m down the track, travels at 11.4 cm/sec.
a) How long does it take for them to collide?
b) Where does the collision occur?

a) sA = vAt sA = 19.5t sB = vBt + so sB = -11.4t + 180 cm note: a negative velocity denotes a velocity in a negative direction. But sA = sB at moment of contact. 19.5t = -11.4t + 180 t = 5.83 sec

b) At collision: sB = sA sB = 19.5t sB = (19.5 cm/s) (5.83 s) sB = 114 cm