Piecewise motion functions

6 A particle \(P\) moves in a straight line. The velocity \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at time \(t \mathrm {~s}\) is given by $$\begin{array} { l l } v = 2 t + 1 & \text { for } 0 \leqslant t \leqslant 5 , \\ v = 36 - t ^ { 2 } & \text { for } 5 \leqslant t \leqslant 7 , \\ v = 2 t - 27 & \text { for } 7 \leqslant t \leqslant 13.5 . \end{array}$$

1. Sketch the velocity-time graph for \(0 \leqslant t \leqslant 13.5\).
2. Find the acceleration at the instant when \(t = 6\).
3. Find the total distance travelled by \(P\) in the interval \(0 \leqslant t \leqslant 13.5\).
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

7 A particle \(P\) moves in a straight line. The velocity \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at time \(t\) seconds is given by $$\begin{array} { l l } v = 0.5 t & \text { for } 0 \leqslant t \leqslant 10 \\ v = 0.25 t ^ { 2 } - 8 t + 60 & \text { for } 10 \leqslant t \leqslant 20 \end{array}$$

1. Show that there is an instantaneous change in the acceleration of the particle at \(t = 10\).
2. Find the total distance covered by \(P\) in the interval \(0 \leqslant t \leqslant 20\).
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

7 A particle moves in a straight line through the point \(O\). The displacement of the particle from \(O\) at time \(t \mathrm {~s}\) is \(s \mathrm {~m}\), where $$\begin{array} { l l } s = t ^ { 2 } - 3 t + 2 & \text { for } 0 \leqslant t \leqslant 6 , \\ s = \frac { 24 } { t } - \frac { t ^ { 2 } } { 4 } + 25 & \text { for } t \geqslant 6 . \end{array}$$

1. Find the value of \(t\) when the particle is instantaneously at rest during the first 6 seconds of its motion.
   At \(t = 6\), the particle hits a barrier at a point \(P\) and rebounds.
2. Find the velocity with which the particle arrives at \(P\) and also the velocity with which the particle leaves \(P\).
3. Find the total distance travelled by the particle in the first 10 seconds of its motion.
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

7 A particle \(P\) travels in a straight line, starting at rest from a point \(O\). The acceleration of \(P\) at time \(t \mathrm {~s}\) after leaving \(O\) is denoted by \(a \mathrm {~m} \mathrm {~s} ^ { - 2 }\), where $$\begin{array} { l l } a = 0.3 t ^ { \frac { 1 } { 2 } } & \text { for } 0 \leqslant t \leqslant 4 , \\ a = - k t ^ { - \frac { 3 } { 2 } } & \text { for } 4 < t \leqslant T , \end{array}$$ where \(k\) and \(T\) are constants.

1. Find the velocity of \(P\) at \(t = 4\).
2. It is given that there is no change in the velocity of \(P\) at \(t = 4\) and that the velocity of \(P\) at \(t = 16\) is \(0.3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Show that \(k = 2.6\) and find an expression, in terms of \(t\), for the velocity of \(P\) for \(4 \leqslant t \leqslant T\).
3. Given that \(P\) comes to instantaneous rest at \(t = T\), find the exact value of \(T\).
4. Find the total distance travelled between \(t = 0\) and \(t = T\).
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

7 A particle moves in a straight line starting from a point \(O\) before coming to instantaneous rest at a point \(X\). At time \(t \mathrm {~s}\) after leaving \(O\), the velocity \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) of the particle is given by $$\begin{array} { l l } v = 7.2 t ^ { 2 } & 0 \leqslant t \leqslant 2 , \\ v = 30.6 - 0.9 t & 2 \leqslant t \leqslant 8 , \\ v = \frac { 1600 } { t ^ { 2 } } + k t & 8 \leqslant t , \end{array}$$ where \(k\) is a constant. It is given that there is no instantaneous change in velocity at \(t = 8\).
Find the distance \(O X\).
If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

6 A particle, \(P\), travels in a straight line, starting from a point \(O\) with velocity \(6 \mathrm {~ms} ^ { - 1 }\). The acceleration of \(P\) at time \(t \mathrm {~s}\) after leaving \(O\) is \(a \mathrm {~m} \mathrm {~s} ^ { - 2 }\), where $$\begin{array} { l l } a = - 1.5 t ^ { \frac { 1 } { 2 } } & \text { for } 0 \leqslant t \leqslant 1 , \\ a = 1.5 t ^ { \frac { 1 } { 2 } } - 3 t ^ { - \frac { 1 } { 2 } } & \text { for } t > 1 . \end{array}$$

1. Find the velocity of \(P\) at \(t = 1\).
2. Given that there is no change in the velocity of \(P\) when \(t = 1\), find an expression for the velocity of \(P\) for \(t > 1\).
   \includegraphics[max width=\textwidth, alt={}, center]{145d93bd-7f56-4e8c-a646-938330511347-11_2725_35_99_20}
3. Given that the velocity of \(P\) is positive for \(t \leqslant 4\), find the total distance travelled between \(t = 0\) and \(t = 4\).
   \includegraphics[max width=\textwidth, alt={}, center]{145d93bd-7f56-4e8c-a646-938330511347-12_723_762_248_653} Two particles, \(A\) and \(B\), of masses 0.2 kg and 0.3 kg respectively, are attached to the ends of a light inextensible string. The string passes over a small fixed smooth pulley which is attached to the bottom of a rough plane inclined at an angle \(\theta\) to the horizontal where \(\sin \theta = 0.6\). Particle \(A\) lies on the plane, and particle \(B\) hangs vertically below the pulley, 0.25 m above horizontal ground. The string between \(A\) and the pulley is parallel to a line of greatest slope of the plane (see diagram). The coefficient of friction between \(A\) and the plane is 1.125 . Particle \(A\) is released from rest.
4. Find the tension in the string and the magnitude of the acceleration of the particles.
   \includegraphics[max width=\textwidth, alt={}, center]{145d93bd-7f56-4e8c-a646-938330511347-12_2716_38_109_2012}
5. When \(B\) reaches the ground, it comes to rest. Find the total distance that \(A\) travels down the plane from when it is released until it comes to rest. You may assume that \(A\) does not reach the pulley.
   If you use the following page to complete the answer to any question, the question number must be clearly shown.
   \includegraphics[max width=\textwidth, alt={}, center]{145d93bd-7f56-4e8c-a646-938330511347-14_2715_31_106_2016}

6 A \(\boldsymbol { p }\) rticle \(P \mathrm {~m}\) s ira straitg lie . Tb ± lo ity \(v \mathrm {~ms} ^ { - 1 }\) at time \(t \mathrm {~s}\) is g ஓ ity $$\begin{array} { l l } v = 5 t ( t - 2 & \text { fo } 0 \leqslant t \leqslant 4 \\ v = k & \text { fo } 4 \leqslant t \leqslant 4 \\ v = 82 \quad t & \text { fo } 4 \leqslant t \leqslant \Omega \end{array}$$ wh re \(k\) is a co tan.

1. Fid \(k\).
2. Sk tcht b lo ity ime g aff \(\mathbf { 0 } \quad 0 \leqslant t \leqslant 0\)
3. Fid bet \(\mathbf { 6 }\) le \(\mathrm { s } \mathbf { 6 } t\) fo wh cht b acceleratio \(P\) is \(\mathbf { p }\) itie .
4. Fid \(\mathbf { b }\) to ald stan e trac lledy \(P\) irt \(\mathbf { b }\) in era \(10 \leqslant t \leqslant 0\)

7 A vehicle is moving in a straight line. The velocity \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at time \(t \mathrm {~s}\) after the vehicle starts is given by $$\begin{aligned} & v = A \left( t - 0.05 t ^ { 2 } \right) \quad \text { for } 0 \leqslant t \leqslant 15 , \\ & v = \frac { B } { t ^ { 2 } } \quad \text { for } t \geqslant 15 , \end{aligned}$$ where \(A\) and \(B\) are constants. The distance travelled by the vehicle between \(t = 0\) and \(t = 15\) is 225 m .

1. Find the value of \(A\) and show that \(B = 3375\).
2. Find an expression in terms of \(t\) for the total distance travelled by the vehicle when \(t \geqslant 15\).
3. Find the speed of the vehicle when it has travelled a total distance of 315 m . \footnotetext{Permission to reproduce items where third-party owned material protected by copyright is included has been sought and cleared where possible. Every reasonable effort has been made by the publisher (UCLES) to trace copyright holders, but if any items requiring clearance have unwittingly been included, the publisher will be pleased to make amends at the earliest possible opportunity. University of Cambridge International Examinations is part of the Cambridge Assessment Group. Cambridge Assessment is the brand name of University of Cambridge Local Examinations Syndicate (UCLES), which is itself a department of the University of Cambridge. }

7 A particle \(P\) moves in a straight line starting from a point \(O\). The velocity \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) of \(P\) at time \(t \mathrm {~s}\) is given by $$\begin{array} { l l } v = 12 t - 4 t ^ { 2 } & \text { for } 0 \leqslant t \leqslant 2 , \\ v = 16 - 4 t & \text { for } 2 \leqslant t \leqslant 4 . \end{array}$$

1. Find the maximum velocity of \(P\) during the first 2 s .
2. Determine, with justification, whether there is any instantaneous change in the acceleration of \(P\) when \(t = 2\).
3. Sketch the velocity-time graph for \(0 \leqslant t \leqslant 4\).
   \includegraphics[max width=\textwidth, alt={}, center]{03e325b9-171a-4f76-95cd-57dad3741caf-13_684_1054_351_584}
4. Find the distance travelled by \(P\) in the interval \(0 \leqslant t \leqslant 4\).
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

6 A particle \(P\) starts from rest at \(O\) and travels in a straight line. Its velocity \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at time \(t \mathrm {~s}\) is given by \(v = 8 t - 2 t ^ { 2 }\) for \(0 \leqslant t \leqslant 3\), and \(v = \frac { 54 } { t ^ { 2 } }\) for \(t > 3\). Find

1. the distance travelled by \(P\) in the first 3 seconds,
2. an expression in terms of \(t\) for the displacement of \(P\) from \(O\), valid for \(t > 3\),
3. the value of \(v\) when the displacement of \(P\) from \(O\) is 27 m .

6 A particle travels along a straight line. It starts from rest at a point \(A\) on the line and comes to rest again, 10 seconds later, at another point \(B\) on the line. The velocity \(t\) seconds after leaving \(A\) is $$\begin{array} { r l l } 0.72 t ^ { 2 } - 0.096 t ^ { 3 } & \text { for } & 0 \leqslant t \leqslant 5 \\ 2.4 t - 0.24 t ^ { 2 } & \text { for } & 5 \leqslant t \leqslant 10 \end{array}$$

1. Show that there is no instantaneous change in the acceleration of the particle when \(t = 5\).
2. Find the distance \(A B\).

7 A vehicle starts from rest at a point \(O\) and moves in a straight line. Its speed \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at time \(t\) seconds after leaving \(O\) is defined as follows. $$\begin{aligned} \text { For } 0 & \leqslant t \leqslant 60 , \quad v = k _ { 1 } t - 0.005 t ^ { 2 } \\ \text { for } t \geqslant 60 , \quad v & = \frac { k _ { 2 } } { \sqrt { } t } \end{aligned}$$ The distance travelled by the vehicle during the first 60 s is 540 m .

1. Find the value of the constant \(k _ { 1 }\) and show that \(k _ { 2 } = 12 \sqrt { } ( 60 )\).
2. Find an expression in terms of \(t\) for the total distance travelled when \(t \geqslant 60\).
3. Find the speed of the vehicle when it has travelled a total distance of 1260 m .

7 A racing car is moving in a straight line. The acceleration \(a \mathrm {~m} \mathrm {~s} ^ { - 2 }\) at time \(t \mathrm {~s}\) after the car starts from rest is given by $$\begin{array} { l l } a = 15 t - 3 t ^ { 2 } & \text { for } 0 \leqslant t \leqslant 5 \\ a = - \frac { 625 } { t ^ { 2 } } & \text { for } 5 < t \leqslant k \end{array}$$ where \(k\) is a constant.

1. Find the maximum acceleration of the car in the first five seconds of its motion.
2. Find the distance of the car from its starting point when \(t = 5\).
3. The car comes to rest when \(t = k\). Find the value of \(k\).

5 A particle starts from a point \(O\) and moves in a straight line. The velocity of the particle at time \(t \mathrm {~s}\) after leaving \(O\) is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), where $$\begin{array} { l l } v = 1.5 + 0.4 t & \text { for } 0 \leqslant t \leqslant 5 , \\ v = \frac { 100 } { t ^ { 2 } } - 0.1 t & \text { for } t \geqslant 5 . \end{array}$$

1. Find the acceleration of the particle during the first 5 seconds of motion.
2. Find the value of \(t\) when the particle is instantaneously at rest.
3. Find the total distance travelled by the particle in the first 10 seconds of motion.

3 \(4 t \quad t t t v t \quad t\)
\(t v\) tt \(t t\) tttvt \(t \quad t \quad t \quad t t \quad t t\)

1. \(x t t \quad t t \quad\) t \(t\) - • • •
2. \(t v\) t \(t\)
3. \(t v\) t \(t v\)
   \(5 t v t v v t\)
4. \(t\) t \(t\)
5. \(t t\)

4 A cyclist travels along a straight road. Her velocity \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), at time \(t\) seconds after starting from a point \(O\), is given by $$\begin{aligned} & v = 2 \quad \text { for } 0 \leqslant t \leqslant 10 \\ & v = 0.03 t ^ { 2 } - 0.3 t + 2 \quad \text { for } t \geqslant 10 . \end{aligned}$$

1. Find the displacement of the cyclist from \(O\) when \(t = 10\).
2. Show that, for \(t \geqslant 10\), the displacement of the cyclist from \(O\) is given by the expression \(0.01 t ^ { 3 } - 0.15 t ^ { 2 } + 2 t + 5\).
3. Find the time when the acceleration of the cyclist is \(0.6 \mathrm {~m} \mathrm {~s} ^ { - 2 }\). Hence find the displacement of the cyclist from \(O\) when her acceleration is \(0.6 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).

1. In this question you must show all stages of your working. Solutions relying on calculator technology are not acceptable.

A particle \(P\) is moving in a straight line.
At time \(t\) seconds, the speed, \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), of \(P\) is given by the continuous function $$v = \begin{cases} \sqrt { 2 t + 1 } & 0 \leqslant t \leqslant k \\ \frac { 3 } { 4 } t & t > k \end{cases}$$ where \(k\) is a constant.

1. Show that \(k = 4\), explaining your method carefully.
2. Find the acceleration of \(P\) when \(t = 1.5\) At time \(t = 0 , P\) passes through the point \(O\)
3. Find the distance of \(P\) from \(O\) when \(t = 8\)

4. A particle \(P\) moves along the \(x\)-axis in a straight line so that, at time \(t\) seconds, the velocity of \(P\) is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), where $$v = \begin{cases} 10 t - 2 t ^ { 2 } , & 0 \leqslant t \leqslant 6 \\ \frac { - 432 } { t ^ { 2 } } , & t > 6 \end{cases}$$ At \(t = 0 , P\) is at the origin \(O\). Find the displacement of \(P\) from \(O\) when

1. \(t = 6\),
2. \(t = 10\).

2. A particle \(P\) moves in a straight line so that, at time \(t\) seconds, its acceleration \(a \mathrm {~m} \mathrm {~s} ^ { - 2 }\) is given by $$a = \begin{cases} 4 t - t ^ { 2 } , & 0 \leq t \leq 3 , \\ \frac { 27 } { t ^ { 2 } } , & t > 3 . \end{cases}$$ At \(t = 0 , P\) is at rest. Find the speed of \(P\) when

1. \(t = 3\),
2. \(t = 6\). \begin{figure}[h]
   \captionsetup{labelformat=empty} \caption{Figure 1} \includegraphics[alt={},max width=\textwidth]{a89db1ee-073c-43a3-8480-44970e51c6e2-3_329_1198_391_515}
   \end{figure} Figure 1 shows the path taken by a cyclist in travelling on a section of a road. When the cyclist comes to the point \(A\) on the top of a hill, she is travelling at \(8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). She descends a vertical distance of 20 m to the bottom of the hill. The road then rises to the point \(B\) through a vertical distance of 12 m . When she reaches \(B\), her speed is \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The total mass of the cyclist and the cycle is 80 kg and the total distance along the road from \(A\) to \(B\) is 500 m . By modelling the resistance to the motion of the cyclist as of constant magnitude 20 N ,
3. find the work done by the cyclist in moving from \(A\) to \(B\). At \(B\) the road is horizontal. Given that at \(B\) the cyclist is accelerating at \(0.5 \mathrm {~m} \mathrm {~s} ^ { - 2 }\),
4. find the power generated by the cyclist at \(B\).

1. A particle \(P\) moves on the \(x\)-axis. At time \(t\) seconds the velocity of \(P\) is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) in the direction of \(x\) increasing, where \(v\) is given by

$$v = \left\{ \begin{array} { l c } 8 t - \frac { 3 } { 2 } t ^ { 2 } , & 0 \leqslant t \leqslant 4 , \\ 16 - 2 t , & t > 4 . \end{array} \right.$$ When \(t = 0 , P\) is at the origin \(O\).
Find

1. the greatest speed of \(P\) in the interval \(0 \leqslant t \leqslant 4\),
2. the distance of \(P\) from \(O\) when \(t = 4\),
3. the time at which \(P\) is instantaneously at rest for \(t > 4\),
4. the total distance travelled by \(P\) in the first 10 s of its motion.

7
\includegraphics[max width=\textwidth, alt={}, center]{57725055-7bce-4ad0-bb1c-59d07d56e2bd-4_531_1481_1194_331} A sprinter \(S\) starts from rest at time \(t = 0\), where \(t\) is in seconds, and runs in a straight line. For \(0 \leqslant t \leqslant 3 , S\) has velocity \(\left( 6 t - t ^ { 2 } \right) \mathrm { m } \mathrm { s } ^ { - 1 }\). For \(3 < t \leqslant 22 , S\) runs at a constant speed of \(9 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). For \(t > 22 , S\) decelerates at \(0.6 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) (see diagram).

1. Express the acceleration of \(S\) during the first 3 seconds in terms of \(t\).
2. Show that \(S\) runs 18 m in the first 3 seconds of motion.
3. Calculate the time \(S\) takes to run 100 m .
4. Calculate the time \(S\) takes to run 200 m . OCR is committed to seeking permission to reproduce all third-party content that it uses in its assessment materials. OCR has attempted to identify and contact all copyright holders whose work is used in this paper. To avoid the issue of disclosure of answer-related information to candidates, all copyright acknowledgements are reproduced in the OCR Copyright Acknowledgements Booklet. This is produced for each series of examinations, is given to all schools that receive assessment material and is freely available to download from our public website (\href{http://www.ocr.org.uk}{www.ocr.org.uk}) after the live examination series.
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2 A box of emergency supplies is dropped to victims of a natural disaster from a stationary helicopter at a height of 1000 metres. The initial velocity of the box is zero. At time \(t \mathrm {~s}\) after being dropped, the acceleration, \(a \mathrm {~m} \mathrm {~s} ^ { - 2 }\), of the box in the vertically downwards direction is modelled by $$\begin{aligned} & a = 10 - t \text { for } 0 \leqslant t \leqslant 10 \\ & a = 0 \quad \text { for } \quad t > 10 \end{aligned}$$

1. Find an expression for the velocity, \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), of the box in the vertically downwards direction in terms of \(t\) for \(0 \leqslant t \leqslant 10\). Show that for \(t > 10 , v = 50\).
2. Draw a sketch graph of \(v\) against \(t\) for \(0 \leqslant t \leqslant 20\).
3. Show that the height, \(h \mathrm {~m}\), of the box above the ground at time \(t\) s is given, for \(0 \leqslant t \leqslant 10\), by $$h = 1000 - 5 t ^ { 2 } + \frac { 1 } { 6 } t ^ { 3 }$$ Find the height of the box when \(t = 10\).
4. Find the value of \(t\) when the box hits the ground.
5. Some of the supplies in the box are damaged when the box hits the ground. So measures are considered to reduce the speed with which the box hits the ground the next time one is dropped. Two different proposals are made. Carry out suitable calculations and then comment on each of them.
   (A) The box should be dropped from a height of 500 m instead of 1000 m .
   (B) The box should be fitted with a parachute so that its acceleration is given by $$\begin{gathered} \quad a = 10 - 2 t \text { for } 0 \leqslant t \leqslant 5 , \\ a = 0 \quad \text { for } \quad t > 5 . \end{gathered}$$

5 A box of emergency supplies is dropped to victims of a natural disaster from a stationary helicopter at a height of 1000 metres. The initial velocity of the box is zero. At time \(t \mathrm {~s}\) after being dropped, the acceleration, \(a \mathrm {~ms} ^ { - 2 }\), of the box in the vertically downwards direction is modelled by $$\begin{aligned} & a = 10 - t \text { for } 0 \leqslant t \leqslant 10 \\ & a = 0 \quad \text { for } \quad t > 10 \end{aligned}$$

1. Find an expression for the velocity, \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), of the box in the vertically downwards direction in terms of \(t\) for \(0 \leqslant t \leqslant 10\). Show that for \(t > 10 , v = 50\).
2. Draw a sketch graph of \(v\) against \(t\) for \(0 \leqslant t \leqslant 20\).
3. Show that the height, \(h \mathrm {~m}\), of the box above the ground at time \(t \mathrm {~s}\) is given, for \(0 \leqslant t \leqslant 10\), by $$h = 1000 - 5 t ^ { 2 } + \frac { 1 } { 6 } t ^ { 3 }$$ Find the height of the box when \(t = 10\).
4. Find the value of \(t\) when the box hits the ground.
5. Some of the supplies in the box are damaged when the box hits the ground. So measures are considered to reduce the speed with which the box hits the ground the next time one is dropped. Two different proposals are made. Carry out suitable calculations and then comment on each of them.
   (A) The box should be dropped from a height of 500 m instead of 1000 m .
   (B) The box should be fitted with a parachute so that its acceleration is given by $$\begin{gathered} \quad a = 10 - 2 t \text { for } 0 \leqslant t \leqslant 5 , \\ a = 0 \quad \text { for } \quad t > 5 . \end{gathered}$$ \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{578555c7-e316-47d3-876a-0b6accce8946-5_342_979_319_633} \captionsetup{labelformat=empty} \caption{Fig. 4}
   \end{figure} Particles P and Q move in the same straight line. Particle P starts from rest and has a constant acceleration towards \(Q\) of \(0.5 \mathrm {~m} \mathrm {~s} ^ { - 2 }\). Particle \(Q\) starts 125 m from \(P\) at the same time and has a constant speed of \(10 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) away from \(P\). The initial values are shown in Fig. 4.
6. Write down expressions for the distances travelled by P and by Q at time \(t\) seconds after the start of the motion.
7. How much time does it take for P to catch up with Q and how far does P travel in this time?