Question

chapter 2. problems 25,26 and 27

Center OI MISS A TOW US PUSSIDIOBANUL ON a problemi y Walter van B. Roberts and Martin Gardner 25 (a) In a race, you run the first half of the distance at speed 14, and the second half at speed u. Find the over-all speed, i.e., the total distance divided by the total time. (b) Check the units of your equation using the method shown in example 1 on p. 22. (c) Check that your answer makes sense in the case where u = v. (d) Show that the dependence of your result on u and w makes sense. That is, first check whether making u bigger makes the result bigger, or smaller. Then compare this with what you expect physically. Problem by B. Shotwell.] Velocity and Relative Motion Chapter 2 26 (a) Let R be the radius of the Earth and T the time (one day) that it takes for one rotation. Find the speed at which a point on the equator moves due to the rotation of the earth. (b) Check the units of your equation using the method shown in example 1 on p. 22. (c) Check that your answer to part a makes sense in the case where the Earth stops rotating completely, so that T is infinitely long. (d) Nairobi, Kenya, is very close to the equator. Plugging in num- bers to your answer from part a, find Nairobi's speed in meters per second. See the table in the back of the book for the relevant data. For comparison, the speed of sound is about 340 m/s. 27 (a) Let O be the latitude of a point on the Earth's surface. Derive an algebra equation for the distance, L. traveled by that point during one rotation of the Earth about its axis, i.e., over one day, expressed in terms of O and R. the radius of the earth. You may find it helpful to draw one or more diagrams in the style of figure h on p. 29. (b) Generalize the result of problem 26a to points not necessarily on the equator. (c) Check the units of your equation using the method shown in example 1 on p. 22. (d) Check that your equation in part b gives zero for the North Pole, and also that it agrees with problem 26a in the special case of a point on the equator. (e) At what speed is Fullerton, California, at latitude 8 = 34°, moving with the rotation of the Earth about its axis? 28 In running races at distances of 800 meters and longer, run-

Answer 1

25

a)

let the total distance be "2x"

u = speed for first half of distance "x"

v = speed for second half of distance "x"

t₁ = time taken for the first half of distance = x/u

t₂ = time taken for the second half of distance = x/v

t = total time = t₁ + t₂ = x/u + x/v

Overall speed is given as

V_o = total distance/total time = 2x/(t₁ + t₂ ) = 2x/(x/u + x/v)

V_o = 2uv/(u + v)

b)

units of "u" = m/s

units of "v" = m/s

V_o = 2uv/(u + v)

V_o = 2 (m/s) (m/s)/((m/s) + (m/s))

V_o = m/s

c)

when u = v

Overall speed is given as

V_o = 2uv/(u + v)

V_o = 2uu/(u + u)

V_o = u m/s

d)

when u > v

Overall speed is given as

V_o = 2uv/(u + v)

V_o = 2u/((u/v) + 1)

u>v is greater than 1

So V_o < 2

l