Simple Speed Comparison Problem

Question

Dale starts walking to school at a constant speed. Five minutes after he leaves, Dale's sister Ruby realizes that Dale forgot his lunch. Ruby knows that she can run two times the speed that Dale is walking, or bike three times his speed. However, her bike is locked in the garage and it takes her two minutes to unlock the bike and bring it outside to ride. Should Ruby run or bike to reach Dale faster?

Answer 1

Dale moves at speed v for a time t to reach the intercept point d = vt

 

t = d/v

 

Ruby walking:  Ruby moves at speed 2v to reach d, but she started 5 minutes late, so she reaches d at time:

 

t1 = 5 + d/(2v) minutes

 

Ruby on bicycle:  Ruby moves at speed 3v to reach d, but started 5 minutes late plus an extra 2 minutes late, so she reaches d at time:

 

t2 = (5+2) + d/(3v) = 7 + d/(3v)

 

Now find out how far Dale moves over the times t1 and t2, and pick the option in which he moves the smaller distance.

 

Ruby walking:

 

v*t1 = d1

 

v*[5 + d1/(2v)] = d1

 

5v + (1/2)*d1 = d1

 

d1 = 10v

 

Ruby bicycling:

 

v*t2 = d2

 

v*[7 + d2/(3v)] = d2

 

7v + (1/3)*d2 = d2

 

d2  = (21/2)*v

 

Compare d1 and d2, and see that d1 < d2, so Ruby intercepts Dale sooner when running than when bicycling.