a) We have that $a=\\var{a}, u=\\var{u}$ and $s=\\var{s}$. Time is unknown so we use a formula without $t$, this is $v^2=u^2+2as$.
\nTherefore
\n\\begin{align} v &= \\sqrt{u^2+2as}, \\\\
&= \\sqrt{\\var{u}^2+\\left(2 \\times \\var{a} \\times \\var{s}\\right)}, \\\\
&= \\sqrt{\\var[fractionNumbers]{u^2+2*a*s}}\\mathrm{ms^{-1}}. \\end{align}
So the speed when she reaches the shop is $\\sqrt{\\var[fractionNumbers]{u^2+2*a*s}} \\mathrm{ms^{-1}}$.
\nb) We have been given $a,u$ and $s$. Therefore to calculate time, $t$, we should use the equation $s=ut+\\frac{1}{2}at^2$ rearranged for $t$. We could then check our answer using $v=u+at$ rearranged for $t$, with the value of $v$ we found in part a), however if we have made a mistake in the previous part this will affect our answer.
\nTherefore we have $\\frac{1}{2}at^2 + ut - s = 0$, so to solve for $t$ we can use the quadratic formula $\\frac{-b+\\sqrt{b^2-4ac}}{2a}$, with $b=u, a=\\frac{1}{2}a$ and $c=-s$. Note that we only take the positive root as we are trying to find time.
\nThis becomes
\n$t = \\frac{-u + \\sqrt{u^2 - \\left(4 \\times \\frac{1}{2}a \\times -s\\right)}}{2 \\times \\frac{1}{2}a}.$
\nTherefore
\n$t = \\frac{-\\var{u} + \\sqrt{\\var{u}^2 + \\left(2\\times \\var{a} \\times \\var{s}\\right)}}{\\var{a}} = \\var{precround((-u+sqrt(u^2+2a*s))/a,3)} \\mathrm{s}$ which has been evaluated to 3d.p.
\nSo the time taken to reach the shop is $\\var{precround((-u+sqrt(u^2+2a*s))/a,3)} \\mathrm{s}.$
\nWe can check this using
\n\\begin{align} v & = u + at, \\\\
& = \\var{u} + \\var{a} \\times \\var{precround((-u+sqrt(u^2+2a*s))/a,3)}, \\\\
& = \\var{u + a*precround((-u+sqrt(u^2+2a*s))/a,3)}, \\\\
& = \\sqrt{\\var[fractionNumbers]{u^2+2*a*s}}\\mathrm{ms^{-1}}. \\end{align}
which is the speed we found in part a).
", "rulesets": {}, "parts": [{"vsetrangepoints": 5, "prompt": "
What is the speed in $\\mathrm{ms^{-1}}$ of the cyclist when she reaches the shop? (You can input a square root using the sqrt() function, for example sqrt(2) for $\\sqrt{2}$)
How long in seconds does it take the cyclist to reach the shop? (Answer to 3d.p.)
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