Two cars traveling at constant speed on a track are side by side every
56 minutes. If, with the same speeds, one of the cars were
traveling in the opposite direction, the two cars would meet
every 8 minutes. How long does it take the faster car to complete
one lap on the track?
A Solution
Several variables come to mind:
vf
= speedoffastcar(units?),
vs
= speedofslowcar(units?),
Tl
= timetakentolap(given),
Tm
= timetakentomeet(given),
Tf
= laptimeforthefastercar,
Ts
= laptimefortheslowercar,
L
= lengthofonelap.
Using that L
= (speed)×(lap time), we
have the two equations
(vf− vs)·56
= L,
(vf+ vs)·8
= L.
The problem asks to find Tf = [L/(vf)].
I'm worried - I have two equations for three unknowns. Adding
and subtracting 8 times the first and 56 times the
second,
2 ·56·8·vf
= (56+8)L,
2 ·56·8·vs
= (56− 8)L.
Success!
Tf
=
L
vf
=
2 ·56·8
56+8
.
Notes
1.
It seems we could determine the lap time of the slower car,
Ts.
2.
With the given data, we cannot determine the actual speeds
of the two cars. What additional data would enable us to determine
the actual speeds?
3.
If we knew the actual speed of the faster car, e.g., 200 km/hr, could we
determine the speed of the slower car?
4.
The reason I did not simplify the numbers in my solution
was that I wanted develop an algebraic method to solve the
problem for general data, Tl, and Tm.
Algebraic Method
Using that L
= (speed)×(lap time), we
have the two equations
(vf− vs)·Tl
= L,
(vf+ vs)·Tm
= L.
The problem asks to find Tf = [L/(vf)].
I'm worried - I have two equations for three unknowns. Adding
and subtracting Tm times the first and Tl times the
second,
2 ·Tl·Tm·vf
= (Tl+Tm)L,
2 ·Tl·Tm·vs
= (Tl− Tm)L.
Success!
Tf
=
L
vf
=
2 ·Tl·Tm
Tl+Tm
.
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version 3.76. On 30 Oct 2006, 19:55.