Maximum or minimum velocity

6 A particle moves in a straight line. It starts from rest from a fixed point \(O\) on the line. Its velocity at time \(t \mathrm {~s}\) after leaving \(O\) is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), where \(v = t ^ { 2 } - 8 t ^ { \frac { 3 } { 2 } } + 10 t\).

1. Find the displacement of the particle from \(O\) when \(t = 1\).
2. Show that the minimum velocity of the particle is \(- 125 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

4 A cyclist starts from rest at a point \(A\) and travels along a straight road \(A B\), coming to rest at \(B\). The displacement of the cyclist from \(A\) at time \(t \mathrm {~s}\) after the start is \(s \mathrm {~m}\), where $$s = 0.004 \left( 75 t ^ { 2 } - t ^ { 3 } \right)$$

1. Show that the distance \(A B\) is 250 m .
2. Find the maximum velocity of the cyclist.

7 A particle \(X\) travels in a straight line. The velocity of \(X\) at time \(t\) s after leaving a fixed point \(O\) is denoted by \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), where $$v = - 0.1 t ^ { 3 } + 1.8 t ^ { 2 } - 6 t + 5.6$$ The acceleration of \(X\) is zero at \(t = p\) and \(t = q\), where \(p < q\).

1. Find the value of \(p\) and the value of \(q\).
   It is given that the velocity of \(X\) is zero at \(t = 14\).
2. Find the velocities of \(X\) at \(t = p\) and at \(t = q\), and hence sketch the velocity-time graph for the motion of \(X\) for \(0 \leqslant t \leqslant 15\).
3. Find the total distance travelled by \(X\) between \(t = 0\) and \(t = 15\).
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

3 A particle \(P\) travels from a point \(O\) along a straight line and comes to instantaneous rest at a point \(A\). The velocity of \(P\) at time \(t \mathrm {~s}\) after leaving \(O\) is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), where \(v = 0.027 \left( 10 t ^ { 2 } - t ^ { 3 } \right)\). Find

1. the distance \(O A\),
2. the maximum velocity of \(P\) while moving from \(O\) to \(A\).

7 A car driver makes a journey in a straight line from \(A\) to \(B\), starting from rest. The speed of the car increases to a maximum, then decreases until the car is at rest at \(B\). The distance travelled by the car \(t\) seconds after leaving \(A\) is \(0.0000117 \left( 400 t ^ { 3 } - 3 t ^ { 4 } \right)\) metres.

1. Find the distance \(A B\).
2. Find the maximum speed of the car.
3. Find the acceleration of the car
   (a) as it starts from \(A\),
   (b) as it arrives at \(B\).
4. Sketch the velocity-time graph for the journey.

7 A particle starts from rest at the point \(A\) and travels in a straight line until it reaches the point \(B\). The velocity of the particle \(t\) seconds after leaving \(A\) is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), where \(v = 0.009 t ^ { 2 } - 0.0001 t ^ { 3 }\). Given that the velocity of the particle when it reaches \(B\) is zero, find

1. the time taken for the particle to travel from \(A\) to \(B\),
2. the distance \(A B\),
3. the maximum velocity of the particle.

7 A motorcyclist starts from rest at \(A\) and travels in a straight line. For the first part of the motion, the motorcyclist's displacement \(x\) metres from \(A\) after \(t\) seconds is given by \(x = 0.6 t ^ { 2 } - 0.004 t ^ { 3 }\).

1. Show that the motorcyclist's acceleration is zero when \(t = 50\) and find the speed \(V \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at this time. For \(t \geqslant 50\), the motorcyclist travels at constant speed \(V \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
2. Find the value of \(t\) for which the motorcyclist's average speed is \(27.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). \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. }

6 A particle \(P\) moves in a straight line, starting from a point \(O\). The velocity of \(P\), measured in \(\mathrm { m } \mathrm { s } ^ { - 1 }\), at time \(t \mathrm {~s}\) after leaving \(O\) is given by $$v = 0.6 t - 0.03 t ^ { 2 }$$

1. Verify that, when \(t = 5\), the particle is 6.25 m from \(O\). Find the acceleration of the particle at this time.
2. Find the values of \(t\) at which the particle is travelling at half of its maximum velocity.

3 A particle \(P\) moves along a straight line for 100 s . It starts at a point \(O\) and at time \(t\) seconds after leaving \(O\) the velocity of \(P\) is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), where $$v = 0.00004 t ^ { 3 } - 0.006 t ^ { 2 } + 0.288 t$$

1. Find the values of \(t\) at which the acceleration of \(P\) is zero.
2. Find the displacement of \(P\) from \(O\) when \(t = 100\).

5 A particle moves in a straight line starting from a point \(O\) with initial velocity \(1 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The acceleration of the particle at time \(t \mathrm {~s}\) after leaving \(O\) is \(a \mathrm {~m} \mathrm {~s} ^ { - 2 }\), where $$a = 1.2 t ^ { \frac { 1 } { 2 } } - 0.6 t$$

1. At time \(T\) s after leaving \(O\) the particle reaches its maximum velocity. Find the value of \(T\).
2. Find the velocity of the particle when its acceleration is maximum (you do not need to verify that the acceleration is a maximum rather than a minimum).

5. A particle moves along the \(x\)-axis. At time \(t\) seconds, \(t \geqslant 0\), the velocity of the particle is \(v \mathrm {~ms} ^ { - 1 }\) in the direction of \(x\) increasing, where \(v = 2 t ^ { \frac { 3 } { 2 } } - 6 t + 2\) At time \(t = 0\) the particle passes through the origin \(O\). At the instant when the acceleration of the particle is zero, the particle is at the point \(A\). Find the distance \(O A\).
(8)

8 The displacement, \(x \mathrm {~m}\), from the origin O of a particle on the \(x\)-axis is given by $$x = 10 + 36 t + 3 t ^ { 2 } - 2 t ^ { 3 }$$ where \(t\) is the time in seconds and \(- 4 \leqslant t \leqslant 6\).

1. Write down the displacement of the particle when \(t = 0\).
2. Find an expression in terms of \(t\) for the velocity, \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), of the particle.
3. Find an expression in terms of \(t\) for the acceleration of the particle.
4. Find the maximum value of \(v\) in the interval \(- 4 \leqslant t \leqslant 6\).
5. Show that \(v = 0\) only when \(t = - 2\) and when \(t = 3\). Find the values of \(x\) at these times.
6. Calculate the distance travelled by the particle from \(t = 0\) to \(t = 4\).
7. Determine how many times the particle passes through O in the interval \(- 4 \leqslant t \leqslant 6\).

5 In a science fiction story a new type of spaceship travels to the moon. The journey takes place along a straight line. The spaceship starts from rest on the earth and arrives at the moon's surface with zero speed. Its speed, \(v\) kilometres per hour at time \(t\) hours after it has started, is given by $$v = 37500 \left( 4 t - t ^ { 2 } \right) .$$

1. Show that the spaceship takes 4 hours to reach the moon.
2. Find an expression for the distance the spaceship has travelled at time \(t\). Hence find the distance to the moon.
3. Find the spaceship's greatest speed during the journey. Section B (36 marks)

2. At time \(t = 0\) a particle \(P\) leaves the origin \(O\) and moves along the \(x\)-axis. At time \(t\) seconds the velocity of \(P\) is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), where $$v = 8 t - t ^ { 2 }$$

1. Find the maximum value of \(v\).
2. Find the time taken for \(P\) to return to \(O\).

6 A swimmer \(C\) swims with velocity \(v \mathrm {~ms} ^ { - 1 }\) in a swimming pool. At time \(t \mathrm {~s}\) after starting, \(v = 0.006 t ^ { 2 } - 0.18 t + k\), where \(k\) is a constant. \(C\) swims from one end of the pool to the other in 28.4 s .

1. Find the acceleration of \(C\) in terms of \(t\).
2. Given that the minimum speed of \(C\) is \(0.65 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), show that \(k = 2\).
3. Express the distance travelled by \(C\) in terms of \(t\), and calculate the length of the pool.

6 A particle \(P\) moves in a straight line. At time \(t\) s after passing through a point \(O\) of the line, the displacement of \(P\) from \(O\) is \(x \mathrm {~m}\). Given that \(x = 0.06 t ^ { 3 } - 0.45 t ^ { 2 } - 0.24 t\), find

1. the velocity and the acceleration of \(P\) when \(t = 0\),
2. the value of \(x\) when \(P\) has its minimum velocity, and the speed of \(P\) at this instant,
3. the positive value of \(t\) when the direction of motion of \(P\) changes.

7 The velocity \(v \mathrm {~ms} ^ { - 1 }\) of a particle at time \(t \mathrm {~s}\) is given by
\(v = 0.5 t ( 7 - t )\). Determine whether the speed of the particle is increasing or decreasing when \(t = 8\).

1 The displacement, \(x \mathrm {~m}\), from the origin O of a particle on the \(x\)-axis is given by $$x = 10 + 36 t + 3 t ^ { 2 } - 2 t ^ { 3 }$$ where \(t\) is the time in seconds and \(- 4 \leqslant t \leqslant 6\).

1. Write down the displacement of the particle when \(t = 0\).
2. Find an expression in terms of \(t\) for the velocity, \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), of the particle.
3. Find an expression in terms of \(t\) for the acceleration of the particle.
4. Find the maximum value of \(v\) in the interval \(- 4 \leqslant t \leqslant 6\).
5. Show that \(v = 0\) only when \(t = - 2\) and when \(t = 3\). Find the values of \(x\) at these times.
6. Calculate the distance travelled by the particle from \(t = 0\) to \(t = 4\).
7. Determine how many times the particle passes through O in the interval \(- 4 \leqslant t \leqslant 6\).

2 \begin{figure}[h] \end{figure} A toy car is moving along the straight line \(\mathrm { O } x\), where O is the origin. The time \(t\) is in seconds. At time \(t = 0\) the car is at \(\mathrm { A } , 3 \mathrm {~m}\) from O as shown in Fig. 5. The velocity of the car, \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), is given by $$v = 2 + 12 t - 3 t ^ { 2 }$$ Calculate the distance of the car from O when its acceleration is zero.

1. A particle \(P\) moves in a straight line so that its velocity \(v \mathrm {~ms} ^ { - 1 }\) at time \(t\) seconds is given, for \(t > 1\), by the formula \(v = 2 t + \frac { 8 } { t ^ { 2 } }\). Find the time when the acceleration of \(P\) is zero. (5 marks)

A key is modelled as a lamina which consists of a circle of radius 3 cm , with a circle of radius 1 cm removed from its centre, attached to a rectangle of length 8 cm and width 1 cm , with a rectangle measuring 3 cm by 1 cm fixed to its end as shown. Calculate the distance of the centre of mass of the key from the line marked \(A B\).

9 A particle \(P\) moves along the \(x\)-axis. At time \(t\) seconds the velocity of \(P\) is \(v \mathrm {~ms} ^ { - 1 }\), where \(v = 2 t ^ { 4 } + k t ^ { 2 } - 4\). The acceleration of \(P\) when \(t = 2\) is \(28 \mathrm {~ms} ^ { - 2 }\).

1. Show that \(k = - 9\).
2. Show that the velocity of \(P\) has its minimum value when \(t = 1.5\). When \(t = 1 , P\) is at the point \(( - 6.4125,0 )\).
3. Find the distance of \(P\) from the origin \(O\) when \(P\) is moving with minimum velocity.
   \includegraphics[max width=\textwidth, alt={}, center]{918c616a-a0c7-4779-8d3c-84ddf1fa36d6-08_698_1009_260_246}
   \(A\) and \(B\) are points at the upper and lower ends, respectively, of a line of greatest slope on a plane inclined at \(30 ^ { \circ }\) to the horizontal. The distance \(A B\) is \(20 \mathrm {~m} . M\) is a point on the plane between \(A\) and \(B\). The surface of the plane is smooth between \(A\) and \(M\), and rough between \(M\) and \(B\). A particle \(P\) is projected with speed \(4.2 \mathrm {~ms} ^ { - 1 }\) from \(A\) down the line of greatest slope (see diagram). \(P\) moves down the plane and reaches \(B\) with speed \(12.6 \mathrm {~ms} ^ { - 1 }\). The coefficient of friction between \(P\) and the rough part of the plane is \(\frac { \sqrt { 3 } } { 6 }\).
4. Find the distance \(A M\).
5. Find the angle between the contact force and the downward direction of the line of greatest slope when \(P\) is in motion between \(M\) and \(B\).

16 An elite athlete runs in a straight line to complete a 100-metre race. During the race, the athlete's velocity, \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), may be modelled by $$v = 11.71 - 11.68 \mathrm { e } ^ { - 0.9 t } - 0.03 \mathrm { e } ^ { 0.3 t }$$ where \(t\) is the time, in seconds, after the starting pistol is fired.
16

1. Find the maximum value of \(v\), giving your answer to one decimal place.
   Fully justify your answer.
   16
2. Find an expression for the distance run in terms of \(t\).
   [0pt] [6 marks]
   16
3. The athlete's actual time for this race is 9.8 seconds. Comment on the accuracy of the model.
   \includegraphics[max width=\textwidth, alt={}, center]{838f0625-95e6-4ad4-b97b-3d3f77cc7f19-27_2488_1719_219_150}