Two cars start together in the same direction from the same place. The first goes with uniform speed of 20 km/h. The second goes at a speed of 12 km/h in the first hour and increases the speed by 1 km/h each succeeding hour. After how many hours will the second car overtake the first car if both cars go non-stop?
Let the second car overtake after ( n ) hours.
Distance by first car: ( 20n ) km.
Distance by second car: speeds ( 12,13,14,\ldots,(11+n) ) form an A.P. [ S_n = \frac{n}{2}\bigl[2\cdot 12 + (n-1)\cdot 1\bigr] = \frac{n}{2}(24 + n - 1) = \frac{n}{2}(n+23). ]
Set equal: [ \frac{n}{2}(n+23) = 20n. ] Assuming ( n\neq 0 ), [ \frac{n+23}{2} = 20 \implies n+23 = 40 \implies n = 17. ]
Answer: 17 hours
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