Standard +0.3 This is a standard M1 two-particle overtaking problem requiring setup of SUVAT equations for both cars, equating positions to find time, then calculating final speed. It's slightly above average difficulty due to the multi-step nature and algebraic manipulation required, but follows a well-practiced procedure with no novel insight needed.
6. Two cars \(A\) and \(B\) are moving in the same direction along a straight horizontal road. Car \(A\) is moving with uniform acceleration \(0.4 \mathrm {~ms} ^ { - 2 }\) and car \(B\) is moving with uniform acceleration \(0.5 \mathrm {~m} \mathrm {~s} ^ { - 2 }\). At the instant when \(B\) is 200 m behind \(A\), the speed of \(A\) is \(35 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and the speed of \(B\) is \(44 \mathrm {~ms} ^ { - 1 }\). Find the speed of \(B\) when it overtakes \(A\).
(9)
M1 for producing a quadratic in \(t\) only from \(s_A =\) their \(s_B \pm 200\)
\(\frac{1}{20}t^2 + 9t - 200 = 0\)
A1
Third A1 for correct '3 term = 0' equation
\((t-20)(t+200) = 0\)
M1
Third M1 for attempt to solve quadratic; must include 200, must be 3 terms, must come from using both distance expressions
\(t = 20\)
A1
Fourth A1 for \(t = 20\)
\(v = 44 + \frac{1}{2}(20) = 54 \text{ ms}^{-1}\)
DM1 A1
DM1 dependent on third M1 for correctly using \(t\) to find \(v\); A1 for 54
N.B. SC for trial and error to find \(t\): can score max M1A1A1M1A0M0A0M1A1 6/9
# Question 6:
| Working/Answer | Marks | Guidance |
|---|---|---|
| $s_A = 35t + \frac{1}{2}(0.4)t^2$; $s_B = 44t + \frac{1}{2}(0.5)t^2$ | M1 A1 A1 | M1 for use of $s = ut + \frac{1}{2}at^2$ for either A or B; first A1 correct equation for A; second A1 correct equation for B |
| $44t + \frac{1}{2}(0.5)t^2 = 200 + 35t + \frac{1}{2}(0.4)t^2$ | M1 | M1 for producing a quadratic in $t$ only from $s_A =$ their $s_B \pm 200$ |
| $\frac{1}{20}t^2 + 9t - 200 = 0$ | A1 | Third A1 for correct '3 term = 0' equation |
| $(t-20)(t+200) = 0$ | M1 | Third M1 for attempt to solve quadratic; must include 200, must be 3 terms, must come from using both distance expressions |
| $t = 20$ | A1 | Fourth A1 for $t = 20$ |
| $v = 44 + \frac{1}{2}(20) = 54 \text{ ms}^{-1}$ | DM1 A1 | DM1 dependent on third M1 for correctly using $t$ to find $v$; A1 for 54 |
**N.B.** SC for trial and error to find $t$: can score max M1A1A1M1A0M0A0M1A1 6/9
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6. Two cars $A$ and $B$ are moving in the same direction along a straight horizontal road. Car $A$ is moving with uniform acceleration $0.4 \mathrm {~ms} ^ { - 2 }$ and car $B$ is moving with uniform acceleration $0.5 \mathrm {~m} \mathrm {~s} ^ { - 2 }$. At the instant when $B$ is 200 m behind $A$, the speed of $A$ is $35 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ and the speed of $B$ is $44 \mathrm {~ms} ^ { - 1 }$. Find the speed of $B$ when it overtakes $A$.\\
(9)\\
\hfill \mbox{\textit{Edexcel M1 2016 Q6 [9]}}