A rocket, speeding along toward Alpha Centauri, has an acceleration a(t) = At²...

^{A rocket, speeding along toward Alpha Centauri, has an acceleration a(t) = At². Assume that the rocket began at rest at the Earth (x = 0) at t = 0. Assuming it simply travels in a straight line from Earth to Alpha Centauri (and beyond), what is the ratio of the speed of the rocket when it has covered half the distance to the star to its speed when it has traveled half the time necessary to reach Alpha Centauri?}

^{a(t) = At²}

^{v(0) = 0}

^{x(0) = 0}

^{D = x(T)}

^{D / 2 = x(τ)}

^{v(τ) / v(½T) = ?}

^Solution

^{a(t) = At²}

^{v(t) = ⅓At³}

^{x(t) = At⁴ / 12}

^{D = AT⁴ / 12}

^{T⁴ = 12D / A}

^{t⁴ = 12x / A}

^{τ⁴ = 12(D / 2) / A}

^{τ⁴ = 12D / A · (½) = (½)T⁴}

^{τ = ⁴√(½) · T}

^{x(τ) = Aτ⁴ / 12 = A(⁴√(½) · T)⁴ / 12 = ½ AT⁴ / 12 = ½D}

^{v(½T) = ⅓A(½T)³ = (½)³ · ⅓AT³}

^{v(τ) = v(⁴√(½) · T) = ⅓A(⁴√(½) · T)³ = (⁴√(½))³ · ⅓AT³}

^{v(τ) / v(½T) = {(⁴√(½))³ · ⅓AT³} / {(½)³ · ⅓AT³} = {(⁴√(½))³ } / {(½)³} = (2 / ⁴√2)³ = 8 / ⁴√8 = (⁴√8)³}