\includegraphics{figure_1} The points \(A\), \(O\) and \(B\) lie on a straight horizontal track as shown in Figure 1. \(A\) is 20 m from \(O\) and \(B\) is on the other side of \(O\) at a distance \(x\) m from \(O\). At time \(t = 0\), a particle \(P\) starts from rest at \(O\) and moves towards \(B\) with uniform acceleration of 3 m s\(^{-2}\). At the same instant, another particle \(Q\), which is at the point \(A\), is moving with a velocity of 3 m s\(^{-1}\) in the direction of \(O\) with uniform acceleration of 4 m s\(^{-2}\) in the same direction. Given that the \(Q\) collides with \(P\) at \(B\), find the value of \(x\). [10 marks]

for $P$: $x = 0 + \frac{3}{2}t^2$ | M1 |
for $Q$: $x + 20 = 3t + 2t^2$ | M1 A1 |
eliminating $x$: $\frac{3}{2}t^2 + 3t - 20 = 0$ | M1 A1 |
$t^2 + 6t - 40 = 0$ i.e. $(t + 10)(t - 4) = 0$ | M1 A1 |
$t = 4$ (only +ve answer) | A1 |
when $t = 4$, $x = \frac{3}{2}(4)^2 = 24$ | M1 A1 | (10)

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\includegraphics{figure_1}

The points $A$, $O$ and $B$ lie on a straight horizontal track as shown in Figure 1. $A$ is 20 m from $O$ and $B$ is on the other side of $O$ at a distance $x$ m from $O$.

At time $t = 0$, a particle $P$ starts from rest at $O$ and moves towards $B$ with uniform acceleration of 3 m s$^{-2}$. At the same instant, another particle $Q$, which is at the point $A$, is moving with a velocity of 3 m s$^{-1}$ in the direction of $O$ with uniform acceleration of 4 m s$^{-2}$ in the same direction.

Given that the $Q$ collides with $P$ at $B$, find the value of $x$. [10 marks]

\hfill \mbox{\textit{Edexcel M1  Q5 [10]}}