Finding constants from motion conditions

6 A particle moves in a straight line \(A B\). The velocity \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) of the particle \(t \mathrm {~s}\) after leaving \(A\) is given by \(v = k \left( t ^ { 2 } - 10 t + 21 \right)\), where \(k\) is a constant. The displacement of the particle from \(A\), in the direction towards \(B\), is 2.85 m when \(t = 3\) and is 2.4 m when \(t = 6\).

1. Find the value of \(k\). Hence find an expression, in terms of \(t\), for the displacement of the particle from \(A\).
2. Find the displacement of the particle from \(A\) when its velocity is a minimum.

7 A particle \(P\) moves in a straight line through a point \(O\). The velocity \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) of \(P\), at time \(t \mathrm {~s}\) after passing \(O\), is given by $$v = \frac { 9 } { 4 } + \frac { b } { ( t + 1 ) ^ { 2 } } - c t ^ { 2 }$$ where \(b\) and \(c\) are positive constants. At \(t = 5\), the velocity of \(P\) is zero and its acceleration is \(- \frac { 13 } { 12 } \mathrm {~m} \mathrm {~s} ^ { - 2 }\).

1. Show that \(b = 9\) and find the value of \(c\).
2. Given that the velocity of \(P\) is zero only at \(t = 5\), find the distance travelled in the first 10 seconds of 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 \(P\) starts at rest and moves in a straight line from a point \(O\). At time \(t\) s after leaving \(O\), the velocity of \(P , v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), is given by \(v = b t + c t ^ { \frac { 3 } { 2 } }\), where \(b\) and \(c\) are constants. \(P\) has velocity \(8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) when \(t = 4\) and has velocity \(13.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) when \(t = 9\).

1. Show that \(b = 3\) and \(c = - 0.5\).
2. Find the acceleration of \(P\) when \(t = 1\).
3. Find the positive value of \(t\) when \(P\) is at instantaneous rest and find the distance of \(P\) from \(O\) at this instant.
4. Find the speed of \(P\) at the instant it returns to \(O\).

6 A particle moves in a straight line, starting from a point \(O\). The velocity of the particle at time \(t\) s after leaving \(O\) is \(v \mathrm {~ms} ^ { - 1 }\). It is given that \(\mathbf { v } = \mathrm { kt } ^ { \frac { 1 } { 2 } } - 2 \mathrm { t } - 8\), where \(k\) is a positive constant. The maximum velocity of the particle is \(4.5 \mathrm {~ms} ^ { - 1 }\).

1. Show that \(k = 10\).
2. 1. Verify that \(v = 0\) when \(t = 1\) and \(t = 16\).
   2. Find the distance travelled by the particle in the first 16 s .

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.

6 A particle \(P\) moves in a straight line passing through a point \(O\). At time \(t \mathrm {~s}\), the velocity of \(P , v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), is given by \(v = q t + r t ^ { 2 }\), where \(q\) and \(r\) are constants. The particle has velocity \(4 \mathrm {~ms} ^ { - 1 }\) when \(t = 1\) and when \(t = 2\).

1. Show that, when \(t = 0.5\), the acceleration of \(P\) is \(4 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).
   …………………………………………………………………………………………......................... .
2. Find the values of \(t\) when \(P\) is at instantaneous rest.
3. The particle is at \(O\) when \(t = 3\). Find the distance of \(P\) from \(O\) when \(t = 0\).

7 A particle moves in a straight line, starting from rest at a point \(O\), and comes to instantaneous rest at a point \(P\). The velocity of the particle at time \(t \mathrm {~s}\) after leaving \(O\) is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), where $$v = 0.6 t ^ { 2 } - 0.12 t ^ { 3 }$$

1. Show that the distance \(O P\) is 6.25 m .
   On another occasion, the particle also moves in the same straight line. On this occasion, the displacement of the particle at time \(t \mathrm {~s}\) after leaving \(O\) is \(s \mathrm {~m}\), where $$s = k t ^ { 3 } + c t ^ { 5 }$$ It is given that the particle passes point \(P\) with velocity \(1.25 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at time \(t = 5\).
2. Find the values of the constants \(k\) and \(c\).
3. Find the acceleration of the particle at time \(t = 5\).
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

6 Particle \(P\) travels in a straight line from \(A\) to \(B\). The velocity of \(P\) at time \(t \mathrm {~s}\) after leaving \(A\) is denoted by \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), where $$v = 0.04 t ^ { 3 } + c t ^ { 2 } + k t$$ \(P\) takes 5 s to travel from \(A\) to \(B\) and it reaches \(B\) with speed \(10 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The distance \(A B\) is 25 m .

1. Find the values of the constants \(c\) and \(k\).
2. Show that the acceleration of \(P\) is a minimum when \(t = 2.5\).

3．A particle \(P\) moves in a horizontal plane．At time \(t\) seconds，the position vector of \(P\) is \(\mathbf { r }\) metres relative to a fixed origin \(O\) ，and \(\mathbf { r }\) is given by $$\mathbf { r } = \left( 18 t - 4 t ^ { 3 } \right) \mathbf { i } + c t ^ { 2 } \mathbf { j } ,$$ where \(c\) is a positive constant．When \(t = 1.5\) ，the speed of \(P\) is \(15 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) ．Find
（a）the value of \(c\) ， （b）the acceleration of \(P\) when \(t = 1.5\) ． \(\mathbf { r }\) metres relative to a fixed origin \(O\) ，and \(\mathbf { r }\) is given by $$\begin{aligned} \mathbf { r } = \left( 18 t - 4 t ^ { 3 } \right) \mathbf { i } + c t ^ { 2 } \mathbf { j } , \\ \text { where } c \text { is a positive constant．When } t = 1.5 \text { ，the speed of } P \text { is } 15 \mathrm {~m} \mathrm {~s} ^ { - 1 } \text { ．Find } \end{aligned}$$ （a）the value of \(c\) ， 3．A particle \(P\) moves in a horizontal plane．At time \(t\) seconds，the position vector of \(P\) is D墐
（b）the acceleration of \(P\) when \(t = 1.5\) ．

3．A particle \(P\) moves in a horizontal plane．At time \(t\) seconds，the position vector of \(P\) is \(\mathbf { r }\) metres relative to a fixed origin \(O\) ，and \(\mathbf { r }\) is given by $$\mathbf { r } = \left( 18 t - 4 t ^ { 3 } \right) \mathbf { i } + c t ^ { 2 } \mathbf { j } ,$$ where \(c\) is a positive constant．When \(t = 1.5\) ，the speed of \(P\) is \(15 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) ．Find
（a）the value of \(c\) ， （b）the acceleration of \(P\) when \(t = 1.5\) ． \(\mathbf { r }\) metres relative to a fixed origin \(O\) ，and \(\mathbf { r }\) is given by $$\begin{aligned} \mathbf { r } = \left( 18 t - 4 t ^ { 3 } \right) \mathbf { i } + c t ^ { 2 } \mathbf { j } , \\ \text { where } c \text { is a positive constant．When } t = 1.5 \text { ，the speed of } P \text { is } 15 \mathrm {~m} \mathrm {~s} ^ { - 1 } \text { ．Find } \end{aligned}$$ （a）the value of \(c\) ， 3．A particle \(P\) moves in a horizontal plane．At time \(t\) seconds，the position vector of \(P\) is D啨
（b）the acceleration of \(P\) when \(t = 1.5\) ．

1. A particle \(P\) moves along a straight line. The speed of \(P\) at time \(t\) seconds ( \(t \geqslant 0\) ) is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), where \(v = \left( p t ^ { 2 } + q t + r \right)\) and \(p , q\) and \(r\) are constants. When \(t = 2\) the speed of \(P\) has its minimum value. When \(t = 0 , v = 11\) and when \(t = 2 , v = 3\)

Find

1. the acceleration of \(P\) when \(t = 3\)
2. the distance travelled by \(P\) in the third second of the motion.
   

4 A particle travels in a straight line. The velocity of the particle at time \(t \mathrm {~s}\) after leaving a point \(O\) is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), where $$v = k t ^ { 2 } - 4 t + 3$$ The distance travelled by the particle in the first 2 s of its motion is 6 m . You may assume that \(v > 0\) in the first 2s of its motion.

1. Find the value of \(k\).
2. Find the value of the minimum velocity of the particle. You do not need to show that this velocity is a minimum.

11 The velocity \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) of a car at time \(t \mathrm {~s}\), during the first 20 s of its journey, is given by \(v = k t + 0.03 t ^ { 2 }\), where \(k\) is a constant. When \(t = 20\) the acceleration of the car is \(1.3 \mathrm {~ms} ^ { - 2 }\). For \(t > 20\) the car continues its journey with constant acceleration \(1.3 \mathrm {~ms} ^ { - 2 }\) until its speed reaches \(25 \mathrm {~ms} ^ { - 1 }\).

1. Find the value of \(k\).
2. Find the total distance the car has travelled when its speed reaches \(25 \mathrm {~ms} ^ { - 1 }\).

12
\includegraphics[max width=\textwidth, alt={}, center]{31b0d5b6-1593-489b-bbcd-486e7c96ff18-09_647_935_260_242} A particle \(P\) moves in a straight line. At time \(t\) seconds, where \(t \geqslant 0\), the velocity of \(P\) is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\). It is given that \(v = - 3 t ^ { 2 } + 24 t + k\), where \(k\) is a positive constant. The diagram shows the velocity-time graph for the motion of \(P\).
\(P\) attains its maximum velocity at time \(T\) seconds. Given that the distance travelled by \(P\) between times \(t = 1\) and \(t = T\) is 297 m , determine the time when \(P\) is instantaneously at rest. \section*{END OF QUESTION PAPER}

7. A rocket is fired from a fixed point \(O\). During the first phase of its motion its velocity, \(v \mathrm { ms } ^ { - 1 }\), is given at time \(t\) seconds after firing by the formula $$v = p t ^ { 2 } + q t .$$ 5 seconds after firing, the rocket is travelling at \(500 \mathrm {~ms} ^ { - 1 }\).
30 seconds after firing, the rocket is travelling at \(12000 \mathrm {~ms} ^ { - 1 }\).

1. Find the constants \(p\) and \(q\).
2. Sketch a velocity-time graph for the rocket for \(0 \leq t \leq 30\).
3. Find the initial acceleration of the rocket.
4. Find the distance of the rocket from \(O 30\) seconds after firing. From time \(t = 30\) onwards, the rocket maintains a constant speed of \(12000 \mathrm {~ms} ^ { - 1 }\).
5. Find the average speed of the rocket during its first 50 seconds of motion.

1. A particle \(P\) moves such that at time \(t\) seconds its position vector, \(\mathbf { r }\) metres, relative to a fixed origin \(O\) is given by

$$\mathbf { r } = \left( \frac { 3 } { 2 } t ^ { 2 } - 3 t \right) \mathbf { i } + \left( \frac { 1 } { 3 } t ^ { 3 } - k t \right) \mathbf { j } ,$$ where \(k\) is a constant and \(\mathbf { i }\) and \(\mathbf { j }\) are perpendicular horizontal unit vectors.

1. Find an expression for the velocity of \(P\) at time \(t\).
2. Given that \(P\) comes to rest instantaneously, find the value of \(k\).

1. A particle \(P\) is moving along the \(x\)-axis.

At time \(t\) seconds, \(t \geqslant 0 , P\) has acceleration \(a \mathrm {~ms} ^ { - 2 }\) and velocity \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) in the direction of \(x\) increasing, where $$v = \mathrm { e } ^ { 2 t } + 6 \mathrm { e } ^ { t } - k t$$ and \(k\) is a positive constant.
When \(t = \ln 2\), \(a = 0\)

1. Find the value of \(k\). When \(t = 0\), the particle passes through the fixed point \(A\).
   When \(t = \ln 2\), the particle is \(d\) metres from \(A\).
2. Showing all stages of your working, find the value of \(d\) correct to 2 significant figures.
   [0pt] [Solutions relying entirely on calculator technology are not acceptable.]

9 In this question you must show detailed reasoning. A car accelerates from rest along a straight level road. The velocity of the car after 8 s is \(25.6 \mathrm {~ms} ^ { - 1 }\).
In one model for the motion, the velocity \(v \mathrm {~ms} ^ { - 1 }\) at time \(t\) seconds is given by \(v = 1.2 t ^ { 2 } - k t ^ { 3 }\), where \(k\) is a constant and \(0 \leqslant t \leqslant 8\).

1. The model gives the correct velocity of \(25.6 \mathrm {~ms} ^ { - 1 }\) at time 8 s . Show that \(k = 0.1\). A second model for the motion uses constant acceleration.
2. Find the value of the acceleration which gives the correct velocity of \(25.6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at time 8 s .
3. Show that these two models give the same value for the displacement in the first 8 s .

14 A particle, P , is moving along a straight line such that its acceleration \(a \mathrm {~ms} ^ { - 2 }\), at any time, \(t\) seconds, may be modelled by $$a = 3 + 0.2 t$$ When \(t = 2\), the velocity of P is \(k \mathrm {~m} \mathrm {~s} ^ { - 1 }\)
14

1. Show that the initial velocity of P is given by the expression \(( k - 6.4 ) \mathrm { ms } ^ { - 1 }\)
   [0pt] [4 marks]
   14
2. The initial velocity of P is one fifth of the velocity when \(t = 2\) Find the value of \(k\).

18 A particle is moving in a straight line through the origin \(O\) The displacement of the particle, \(r\) metres, from \(O\), at time \(t\) seconds is given by $$r = p + 2 t - q \mathrm { e } ^ { - 0.2 t }$$ where \(p\) and \(q\) are constants.
When \(t = 3\), the acceleration of the particle is \(- 1.8 \mathrm {~m} \mathrm {~s} ^ { - 2 }\)
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1. Show that \(q \approx 82\)
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2. The particle has an initial displacement of 5 metres. Find the value of \(p\) Give your answer to two significant figures.
   Turn over for the next question