OCR MEI M1 — Question 3 8 marks

Exam BoardOCR MEI
ModuleM1 (Mechanics 1)
Marks8
PaperDownload PDF ↗
Mark schemeDownload PDF ↗
TopicTravel graphs
TypeTwo-particle meeting or overtaking
DifficultyModerate -0.3 This is a standard M1 kinematics question requiring integration of acceleration to find velocity and displacement, then comparing positions. The piecewise functions add slight complexity, but the calculus is straightforward and the question is highly structured with clear guidance. Slightly easier than average due to the scaffolding and routine techniques.
Spec3.02a Kinematics language: position, displacement, velocity, acceleration3.02b Kinematic graphs: displacement-time and velocity-time3.02c Interpret kinematic graphs: gradient and area3.02f Non-uniform acceleration: using differentiation and integration
Two girls, Marie and Nina, are members of an Olympic hockey team. They are doing fitness training. Marie runs along a straight line at a constant speed of \(6\) ms\(^{-1}\). Nina is stationary at a point O on the line until Marie passes her. Nina immediately runs after Marie until she catches up with her. The time, \(t\) s, is measured from the moment when Nina starts running. So when \(t = 0\), both girls are at O. Nina's acceleration, \(a\) ms\(^{-2}\), is given by \begin{align} a &= 4 - t \quad \text{for } 0 < t < 4,
a &= 0 \quad \text{for } t > 4. \end{align}
  1. Show that Nina's speed, \(v\) ms\(^{-1}\), is given by \begin{align} v &= 4t - \frac{1}{2}t^2 \quad \text{for } 0 < t < 4,
    v &= 8 \quad \text{for } t > 4. \end{align} [3]
  2. Find an expression for the distance Nina has run at time \(t\), for \(0 \leqslant t < 4\). Find how far Nina has run when \(t = 4\) and when \(t = 5\frac{1}{4}\). [4]
  3. Show that Nina catches up with Marie when \(t = 5\frac{1}{4}\). [1]
Question 3:
AnswerMarks Guidance
3(i) v(4-t)dt
1
v4t t2 c (t 0,v0c0)
2
1
v4t t2 for 0t4
2
When t 4,v8 and for t 4, a0 so
AnswerMarks
v8 for t 4M1
A1
B1
AnswerMarks
[3]Attempt to integrate
Condone no mention of arbitrary constant
a0 must be seen or implied
AnswerMarks
(ii)s(4t-1t2)dt
2
1
s2t2  t3
6
When t 4, Nina has travelled
1 1
242  43 21 m
6 3
When t 51, Nina has travelled
3
1
21 811 32m
AnswerMarks
3 3M1
A1
A1
F1
AnswerMarks
[4]Again condone no mention of arbitrary constant
1
Allow follow through from their 21
3
Exact answer required; if rounded to 32, award 0
AnswerMarks
(iii)1
When t 5 , Marie has run 651 32m.
3 3
AnswerMarks
Nina has also run 32 m so caught up MarieB1
[1]1
Allow an equivalent argument that when Marie has run 32 m, t 5 ,
3
as for Nina
This mark is dependent on an answer 32 in part (ii) but allow this
where it is a rounded answer and in this particular case the rounding
can be in part (iii)
AnswerMarks
marknotes 
Question 3:
3 | (i) | v(4-t)dt
1
v4t t2 c (t 0,v0c0)
2
1
v4t t2 for 0t4
2
When t 4,v8 and for t 4, a0 so
v8 for t 4 | M1
A1
B1
[3] | Attempt to integrate
Condone no mention of arbitrary constant
a0 must be seen or implied
(ii) | s(4t-1t2)dt
2
1
s2t2  t3
6
When t 4, Nina has travelled
1 1
242  43 21 m
6 3
When t 51, Nina has travelled
3
1
21 811 32m
3 3 | M1
A1
A1
F1
[4] | Again condone no mention of arbitrary constant
1
Allow follow through from their 21
3
Exact answer required; if rounded to 32, award 0
(iii) | 1
When t 5 , Marie has run 651 32m.
3 3
Nina has also run 32 m so caught up Marie | B1
[1] | 1
Allow an equivalent argument that when Marie has run 32 m, t 5 ,
3
as for Nina
This mark is dependent on an answer 32 in part (ii) but allow this
where it is a rounded answer and in this particular case the rounding
can be in part (iii)
mark | notes
Two girls, Marie and Nina, are members of an Olympic hockey team. They are doing fitness training.

Marie runs along a straight line at a constant speed of $6$ ms$^{-1}$.

Nina is stationary at a point O on the line until Marie passes her. Nina immediately runs after Marie until she catches up with her.

The time, $t$ s, is measured from the moment when Nina starts running. So when $t = 0$, both girls are at O.

Nina's acceleration, $a$ ms$^{-2}$, is given by
\begin{align}
a &= 4 - t \quad \text{for } 0 < t < 4, \\
a &= 0 \quad \text{for } t > 4.
\end{align}

\begin{enumerate}[label=(\roman*)]
\item Show that Nina's speed, $v$ ms$^{-1}$, is given by
\begin{align}
v &= 4t - \frac{1}{2}t^2 \quad \text{for } 0 < t < 4, \\
v &= 8 \quad \text{for } t > 4.
\end{align} [3]
\item Find an expression for the distance Nina has run at time $t$, for $0 \leqslant t < 4$.

Find how far Nina has run when $t = 4$ and when $t = 5\frac{1}{4}$. [4]
\item Show that Nina catches up with Marie when $t = 5\frac{1}{4}$. [1]
\end{enumerate}

\hfill \mbox{\textit{OCR MEI M1  Q3 [8]}}
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