Fly off into space

Classical Mechanics Level pending

A car has an acceleration, a a which depends on distance, s s . If the car's acceleration is a ( s ) = ( s + 2 ) ( s + 3 ) ms − 2 a(s)=(s+2)(s+3)\text{ms}^{-2} , and the car starts from rest, what is its speed after travelling 1 km 1\text{km} , in km/s \text{km/s} ? Give your answer to one decimal place.


The answer is 25.9.

Charley Shi by Charley Shi , 19, New Zealand

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1 solution

a ( s ) = v d v d s = ( s + 2 ) ( s + 3 ) = s 2 + 5 s + 6 , v ( 0 ) = 0 ⟹ v = 2 3 s 3 + 5 s 2 + 12 s a(s)=v\dfrac{dv}{ds}=(s+2)(s+3)=s^2+5s+6, v(0)=0\implies v=\sqrt {\frac{2}{3}s^3+5s^2+12s} .

Substituting s = 1 s=1 km. = 1000 =1000 m. , we get v = 2 3 × 1 0 3 + 5 + 12 × 1 0 − 3 ≈ 25.916 v=\sqrt {\frac{2}{3}\times 10^3+5+12\times 10^{-3}}\approx \boxed {25.916} km/s.

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