Maximum or minimum velocity

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 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 \includegraphics[max width=\textwidth, alt={}, center]{ee138c3f-51e1-4a69-9750-9eb49ac87e22-4_719_1059_264_543} An object \(P\) travels from \(A\) to \(B\) in a time of 80 s . The diagram shows the graph of \(v\) against \(t\), where \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) is the velocity of \(P\) at time \(t \mathrm {~s}\) after leaving \(A\). The graph consists of straight line segments for the intervals \(0 \leqslant t \leqslant 10\) and \(30 \leqslant t \leqslant 80\), and a curved section whose equation is \(v = - 0.01 t ^ { 2 } + 0.5 t - 1\) for \(10 \leqslant t \leqslant 30\). Find

1. the maximum velocity of \(P\),
2. the distance \(A B\).

7 A particle travels in a straight line from \(A\) to \(B\) in 20 s . Its acceleration \(t\) seconds after leaving \(A\) is \(a \mathrm {~m} \mathrm {~s} ^ { - 2 }\), where \(a = \frac { 3 } { 160 } t ^ { 2 } - \frac { 1 } { 800 } t ^ { 3 }\). It is given that the particle comes to rest at \(B\).

1. Show that the initial speed of the particle is zero.
2. Find the maximum speed of the particle.
3. Find the distance \(A B\).

4 A particle \(P\) starts at the point \(O\) and travels in a straight line. At time \(t\) seconds after leaving \(O\) the velocity of \(P\) is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), where \(v = 0.75 t ^ { 2 } - 0.0625 t ^ { 3 }\). Find

1. the positive value of \(t\) for which the acceleration is zero,
2. the distance travelled by \(P\) before it changes its direction of motion.

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
   1. as it starts from \(A\),
   2. as it arrives at \(B\).
   3. 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 \(27.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

7 A particle \(P\) starts from a point \(O\) and moves along a straight line. \(P\) 's velocity \(t\) s after leaving \(O\) is \(\nu \mathrm { m } \mathrm { s } ^ { - 1 }\), where $$v = 0.16 t ^ { \frac { 3 } { 2 } } - 0.016 t ^ { 2 } .$$ \(P\) comes to rest instantaneously at the point \(A\).

1. Verify that the value of \(t\) when \(P\) is at \(A\) is 100 .
2. Find the maximum speed of \(P\) in the interval \(0 < t < 100\).
3. Find the distance \(O A\).
4. Find the value of \(t\) when \(P\) passes through \(O\) on returning from \(A\).

7 A particle \(P\) starts to move from a point \(O\) and travels in a straight line. The velocity of \(P\) is \(k \left( 60 t ^ { 2 } - t ^ { 3 } \right) \mathrm { ms } ^ { - 1 }\) at time \(t \mathrm {~s}\) after leaving \(O\), where \(k\) is a constant. The maximum velocity of \(P\) is \(6.4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

1. Show that \(k = 0.0002\). \(P\) comes to instantaneous rest at a point \(A\) on the line. Find
2. the distance \(O A\),
3. the magnitude of the acceleration of \(P\) at \(A\),
4. the speed of \(P\) when it subsequently passes through \(O\).

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.

7 A particle moves in a straight line. The particle is initially at rest at a point \(O\) on the line. At time \(t \mathrm {~s}\) after leaving \(O\), the acceleration \(a \mathrm {~m} \mathrm {~s} ^ { - 2 }\) of the particle is given by \(a = 25 - t ^ { 2 }\) for \(0 \leqslant t \leqslant 9\).

1. Find the maximum velocity of the particle in this time period.
2. Find the total distance travelled until the maximum velocity is reached.
   The acceleration of the particle for \(t > 9\) is given by \(a = - 3 t ^ { - \frac { 1 } { 2 } }\).
3. Find the velocity of the particle when \(t = 25\).
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

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 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.

3 The velocity-time graph for the motion of a particle is shown below. The velocity \(v \mathrm {~ms} ^ { - 1 }\) at time \(t \mathrm {~s}\) is given by \(\mathrm { v } = - \mathrm { t } ^ { 2 } + 6 \mathrm { t } - 6\) where \(0 \leqslant t \leqslant 5\). \includegraphics[max width=\textwidth, alt={}, center]{7af62e61-c67f-4d05-b6b9-c1a110345812-3_860_979_1082_239}

1. Find the times at which the velocity is \(2 \mathrm {~ms} ^ { - 1 }\).
2. Write down the greatest speed of the particle.

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 \(t = 8\).

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.

10 A particle \(P\) moves in a straight line starting from \(O\). At time \(t\) seconds after leaving \(O\), the velocity of \(P\) is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), where \(v = 5 + 1.5 t - 0.125 t ^ { 3 }\).

1. Find the displacement of \(P\) between the times \(t = 1\) and \(t = 4\).
2. Find the time at which the velocity of \(P\) is a maximum, justifying your answer.

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

1. At time \(T\) s after leaving \(O\) the particle reaches its maximum velocity. Find the value of \(T\). [2]
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). [6]

A particle \(P\) moves along the \(x\)-axis. At time \(t\) seconds the velocity of \(P\) is \(v \text{ ms}^{-1}\) in the positive \(x\)-direction, where \(v = 3t^2 - 4t + 3\). When \(t = 0\), \(P\) is at the origin \(O\). Find the distance of \(P\) from \(O\) when \(P\) is moving with minimum velocity. [8]

A particle \(P\) moves along the \(x\)-axis. At time \(t\) seconds the velocity of \(P\) is \(v\text{m s}^{-1}\), where \(v = 2t^4 + kt^2 - 4\). The acceleration of \(P\) when \(t = 2\) is \(28\text{m s}^{-2}\).

1. Show that \(k = -9\). [3]
2. Show that the velocity of \(P\) has its minimum value when \(t = 1.5\). [3]

When \(t = 1\), \(P\) is at the point \((-6.4125, 0)\).

1. Find the distance of \(P\) from the origin \(O\) when \(P\) is moving with minimum velocity. [4]