Piecewise motion functions

5 A particle starts from rest from a point \(O\) and moves in a straight line. The acceleration of the particle at time \(t \mathrm {~s}\) after leaving \(O\) is \(a \mathrm {~m} \mathrm {~s} ^ { - 2 }\), where \(a = k t ^ { \frac { 1 } { 2 } }\) for \(0 \leqslant t \leqslant 9\) and where \(k\) is a constant. The velocity of the particle at \(t = 9\) is \(1.8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

1. Show that \(k = 0.1\).
   For \(t > 9\), the velocity \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) of the particle is given by \(v = 0.2 ( t - 9 ) ^ { 2 } + 1.8\).
2. Show that the distance travelled in the first 9 seconds is one tenth of the distance travelled between \(t = 9\) and \(t = 18\).
3. Find the greatest acceleration of the particle during the first 10 seconds of its motion.

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.

6 A particle \(P\) moves in a straight line starting from a point \(O\) and comes to rest 14 s later. At time \(t \mathrm {~s}\) after leaving \(O\), the velocity \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) of \(P\) is given by $$\begin{array} { l l } v = p t ^ { 2 } - q t & 0 \leqslant t \leqslant 6 \\ v = 63 - 4.5 t & 6 \leqslant t \leqslant 14 \end{array}$$ where \(p\) and \(q\) are positive constants.
The acceleration of \(P\) is zero when \(t = 2\).

1. Given that there are no instantaneous changes in velocity, find \(p\) and \(q\).
2. Sketch the velocity-time graph.
3. Find the total distance travelled by \(P\) during the 14 s . \includegraphics[max width=\textwidth, alt={}, center]{e1b91e54-a3ae-436c-a4f7-7095891f7034-10_326_1109_255_520} Two particles \(A\) and \(B\) of masses 2 kg and 3 kg respectively are connected by a light inextensible string. Particle \(B\) is on a smooth fixed plane which is at an angle of \(18 ^ { \circ }\) to horizontal ground. The string passes over a fixed smooth pulley at the top of the plane. Particle \(A\) hangs vertically below the pulley and is 0.45 m above the ground (see diagram). The system is released from rest with the string taut. When \(A\) reaches the ground, the string breaks. Find the total distance travelled by \(B\) before coming to instantaneous rest. You may assume that \(B\) does not reach the pulley.
   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.
   1. 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}
   2. 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 particle \(P\) moves in a straight line. The velocity \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at time \(t \mathrm {~s}\) is given by $$\begin{array} { l l } v = 5 t ( t - 2 ) & \text { for } 0 \leqslant t \leqslant 4 \\ v = k & \text { for } 4 \leqslant t \leqslant 14 \\ v = 68 - 2 t & \text { for } 14 \leqslant t \leqslant 20 \end{array}$$ where \(k\) is a constant.

1. Find \(k\).
2. Sketch the velocity-time graph for \(0 \leqslant t \leqslant 20\).
3. Find the set of values of \(t\) for which the acceleration of \(P\) is positive.
4. Find the total distance travelled by \(P\) in the interval \(0 \leqslant t \leqslant 20\).

6 A particle \(P\) of mass 0.6 kg is projected vertically upwards with speed \(5.2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) from a point \(O\) which is 6.2 m above the ground. Air resistance acts on \(P\) so that its deceleration is \(10.4 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) when \(P\) is moving upwards, and its acceleration is \(9.6 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) when \(P\) is moving downwards. Find

1. the greatest height above the ground reached by \(P\),
2. the speed with which \(P\) reaches the ground,
3. the total work done on \(P\) by the air resistance.

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.

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 \(0.6 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).

6 A particle starts from rest at the point A and travels in a straight line. The displacement sm of the particle from A at time ts after leaving A is given by $$s = 0.001 t ^ { 4 } - 0.04 t ^ { 3 } + 0.6 t ^ { 2 } , \quad \text { for } 0 \leqslant t \leqslant 10 .$$

1. Show that the velocity of the particle is \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) when \(\mathrm { t } = 10\). The acceleration of the particle for \(t \geqslant 10\) is \(( 0.8 - 0.08 t ) \mathrm { m } \mathrm { s } ^ { - 2 }\).
2. Show that the velocity of the particle is zero when \(\mathrm { t } = 20\).
3. Find the displacement from A of the particle when \(\mathrm { t } = 20\).

3 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 \(6 \mathrm {~ms} ^ { - 1 }\). 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, \(t \mathrm {~s}\), is measured from the moment when Nina starts running. So when \(t = 0\), both girls are at O .
Nina's acceleration, \(a \mathrm {~ms} ^ { - 2 }\), is given by $$\begin{array} { l l } a = 4 - t & \text { for } 0 \leqslant t \leqslant 4 , \\ a = 0 & \text { for } t > 4 . \end{array}$$

1. Show that Nina's speed, \(v \mathrm {~ms} ^ { - 1 }\), is given by $$\begin{array} { l l } v = 4 t - \frac { 1 } { 2 } t ^ { 2 } & \text { for } 0 \leqslant t \leqslant 4 , \\ v = 8 & \text { for } t > 4 . \end{array}$$
2. Find an expression for the distance Nina has run at time \(t\), for \(0 \leqslant t \leqslant 4\). Find how far Nina has run when \(t = 4\) and when \(t = 5 \frac { 1 } { 3 }\).
3. Show that Nina catches up with Marie when \(t = 5 \frac { 1 } { 3 }\).

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. \(t = 10\).

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.

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 $$\begin{gathered} \quad a = 10 - 2 t \text { for } 0 \leqslant t \leqslant 5 , \\ a = 0 \quad \text { for } \quad t > 5 . \end{gathered}$$

A particle \(P\) moves in a straight line. The velocity \(v\text{ms}^{-1}\) at time \(t\) seconds is given by $$v = 0.5t \quad \text{for } 0 \leqslant t \leqslant 10,$$ $$v = 0.25t^2 - 8t + 60 \quad \text{for } 10 < t \leqslant 20.$$

1. Show that there is an instantaneous change in the acceleration of the particle at \(t = 10\). [3]
2. Find the total distance covered by \(P\) in the interval \(0 \leqslant t \leqslant 20\). [6]

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

1. Find the velocity of \(P\) at \(t = 4\). [2]

1. 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 \(0.3 \text{ m s}^{-1}\). Show that \(k = 2.6\) and find an expression, in terms of \(t\), for the velocity of \(P\) for \(4 \leqslant t \leqslant T\). [4]

1. Given that \(P\) comes to instantaneous rest at \(t = T\), find the exact value of \(T\). [2]

1. Find the total distance travelled between \(t = 0\) and \(t = T\). [4]

A particle \(P\) travels in a straight line from \(A\) to \(D\), passing through the points \(B\) and \(C\). For the section \(AB\) the velocity of the particle is \((0.5t - 0.01t^2)\) m s\(^{-1}\), where \(t\) s is the time after leaving \(A\).

1. Given that the acceleration of \(P\) at \(B\) is 0.1 m s\(^{-2}\), find the time taken for \(P\) to travel from \(A\) to \(B\). [3]

The acceleration of \(P\) from \(B\) to \(C\) is constant and equal to 0.1 m s\(^{-2}\).

1. Given that \(P\) reaches \(C\) with speed 14 m s\(^{-1}\), find the time taken for \(P\) to travel from \(B\) to \(C\). [3]

\(P\) travels with constant deceleration 0.3 m s\(^{-2}\) from \(C\) to \(D\). Given that the distance \(CD\) is 300 m, find

1. the speed with which \(P\) reaches \(D\), [2]
2. the distance \(AD\). [6]

A vehicle is moving in a straight line. The velocity \(v\) m s\(^{-1}\) at time \(t\) s after the vehicle starts is given by $$v = A(t - 0.05t^2) \quad \text{for } 0 \leq t \leq 15,$$ $$v = \frac{B}{t^2} \quad \text{for } t \geq 15,$$ 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 \(B = 3375\). [5]
2. Find an expression in terms of \(t\) for the total distance travelled by the vehicle when \(t \geq 15\). [3]
3. Find the speed of the vehicle when it has travelled a total distance of 315 m. [3]

A vehicle is moving in a straight line. The velocity \(v \text{ m s}^{-1}\) at time \(t \text{ s}\) after the vehicle starts is given by $$v = A(t - 0.05t^2) \text{ for } 0 \leq t \leq 15,$$ $$v = \frac{B}{t} \text{ for } t \geq 15,$$ where \(A\) and \(B\) are constants. The distance travelled by the vehicle between \(t = 0\) and \(t = 15\) is \(225 \text{ m}\).

1. Find the value of \(A\) and show that \(B = 3375\). [5]
2. Find an expression in terms of \(t\) for the total distance travelled by the vehicle when \(t \geq 15\). [3]
3. Find the speed of the vehicle when it has travelled a total distance of \(315 \text{ m}\). [3]

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

1. Find the maximum velocity of \(P\) during the first \(2\text{ s}\). [3]
2. Determine, with justification, whether there is any instantaneous change in the acceleration of \(P\) when \(t = 2\). [2]
3. Sketch the velocity-time graph for \(0 \leqslant t \leqslant 4\). [3]
4. Find the distance travelled by \(P\) in the interval \(0 \leqslant t \leqslant 4\). [5]