Question

In a special race, the goal is to get from point A on a pool’s deck to point B in the water, where there is a buoy, in the shortest amount of time if one is allowed to run or swim.

B. JS. erson

(a) Find an expression for the time an athlete takes to get from A to B, if she is running a distance x with the constant speed vR and then swimming from S to B with the constant speed vS . The expression should only depend on x, dB, dP, vR and vS.

(b) What must x be in order for this time to be minimized?

(c) If the athlete runs an 8 min/mile pace and then swims at a constant speed of 50 yards per 1:40 minutes, what is the value of x that minimizes the crossing time (choose your own values for dB and dP ).

(d) It would be interesting and useful down the line to graph the function you derived at part (a).

Answer 1

(a)SB=sqrt((dB-x)^2+dp^2)

total time = time taken by running + time taken in swimming=(x/vR)+(sqrt((dB-x)^2+dp^2))/vS

$t=(\frac{x}{v_{R}})+\sqrt{(dB-x)^2+d_{p}^2}/v_{S}$

(b)see sqrt value will always be positive so least value possible is

x=dB

then t will be

t=dB/vR+dP/vS

(c)to minimise the time

x=dB