Total distance with direction changes

2 A particle \(P\) moves in a straight line, starting from a point \(O\). At time \(t \mathrm {~s}\) after leaving \(O\), the velocity of \(P , v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), is given by \(v = 4 t ^ { 2 } - 8 t + 3\).

1. Find the two values of \(t\) at which \(P\) is at instantaneous rest.
2. Find the distance travelled by \(P\) between these two times.

7 A particle moves in a straight line starting from rest from a point \(O\). The acceleration of the particle at time \(t \mathrm {~s}\) after leaving \(O\) is \(a \mathrm {~m} \mathrm {~s} ^ { - 2 }\), where $$a = 5.4 - 1.62 t$$

1. Find the positive value of \(t\) at which the velocity of the particle is zero, giving your answer as an exact fraction.
2. Find the velocity of the particle at \(t = 10\) and sketch the velocity-time graph for the first ten seconds of the motion.
3. Find the total distance travelled during the first ten seconds of the 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.

1. A particle \(P\) moves along a straight line. At time \(t = 0 , P\) passes the point \(A\) on the line and at time \(t\) seconds the velocity of \(P\) is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) where

$$v = ( 2 t - 3 ) ( t - 2 )$$ At \(t = 3 , P\) reaches the point \(B\). Find the total distance moved by \(P\) as it travels from \(A\) to \(B\).
(6)

2 A particle moves along the \(x\)-axis with velocity, \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), at time \(t\) given by $$v = 24 t - 6 t ^ { 2 }$$ The positive direction is in the sense of \(x\) increasing.

1. Find an expression for the acceleration of the particle at time \(t\).
2. Find the times, \(t _ { 1 }\) and \(t _ { 2 }\), at which the particle has zero speed.
3. Find the distance travelled between the times \(t _ { 1 }\) and \(t _ { 2 }\).

7 Fig. 7 is a sketch of part of the velocity-time graph for the motion of an insect walking in a straight line. Its velocity, \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), at time \(t\) seconds for the time interval \(- 3 \leqslant t \leqslant 5\) is given by $$v = t ^ { 2 } - 2 t - 8 .$$ \begin{figure}[h] \end{figure}

1. Write down the velocity of the insect when \(t = 0\).
2. Show that the insect is instantaneously at rest when \(t = - 2\) and when \(t = 4\).
3. Determine the velocity of the insect when its acceleration is zero. Write down the coordinates of the point A shown in Fig. 7.
4. Calculate the distance travelled by the insect from \(t = 1\) to \(t = 4\).
5. Write down the distance travelled by the insect in the time interval \(- 2 \leqslant t \leqslant 4\).
6. How far does the insect walk in the time interval \(1 \leqslant t \leqslant 5\) ?

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 \(- 4 \leqslant t \leqslant 6\).

3. A particle \(P\) moves along the \(x\)-axis. At time \(t = 0 , P\) passes through the origin with speed \(6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) in the positive \(x\) direction. The acceleration of \(P\) at time \(t\) seconds, where \(t \geqslant 0\), is \(( 4 t - 8 ) \mathrm { m } \mathrm { s } ^ { - 2 }\) in the positive \(x\) direction.

1. 1. Show that \(P\) is instantaneously at rest when \(t = 1\)
   2. Find the other value of \(t\) for which \(P\) is instantaneously at rest.
2. Find the total distance travelled by \(P\) in the interval \(1 \leqslant t \leqslant 4\)

3. A particle \(P\) moves on the \(x\)-axis. At time \(t = 0 , P\) is instantaneously at rest at \(O\).
At time \(t\) seconds, \(t > 0\), the \(x\) coordinate of \(P\) is given by $$x = 2 t ^ { \frac { 7 } { 2 } } - 14 t ^ { \frac { 5 } { 2 } } + \frac { 56 } { 3 } t ^ { \frac { 3 } { 2 } }$$ Find

1. the non-zero values of \(t\) for which \(P\) is at instantaneous rest
2. the total distance travelled by \(P\) in the interval \(0 \leqslant t \leqslant 4\)
3. the acceleration of \(P\) when \(t = 4\) \(\_\_\_\_\)

3. 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 $$v = 2 t ^ { 2 } - 14 t + 20 , \quad t \geqslant 0$$ Find

1. the times when \(P\) is instantaneously at rest,
2. the greatest speed of \(P\) in the interval \(0 \leqslant t \leqslant 4\)
3. the total distance travelled by \(P\) in the interval \(0 \leqslant t \leqslant 4\)

1. A particle \(P\) moves on the positive \(x\)-axis. The velocity of \(P\) at time \(t\) seconds is \(\left( 2 t ^ { 2 } - 9 t + 4 \right) \mathrm { m } \mathrm { s } ^ { - 1 }\). When \(t = 0 , P\) is 15 m from the origin \(O\).

Find

1. the values of \(t\) when \(P\) is instantaneously at rest,
2. the acceleration of \(P\) when \(t = 5\)
3. the total distance travelled by \(P\) in the interval \(0 \leqslant t \leqslant 5\)

1. 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 \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) in the positive \(x\) direction, where

$$v = 3 t ^ { 2 } - 16 t + 21$$ The particle is instantaneously at rest when \(t = t _ { 1 }\) and when \(t = t _ { 2 } \left( t _ { 1 } < t _ { 2 } \right)\).

1. Find the value of \(t _ { 1 }\) and the value of \(t _ { 2 }\).
2. Find the magnitude of the acceleration of \(P\) at the instant when \(t = t _ { 1 }\).
3. Find the distance travelled by \(P\) in the interval \(t _ { 1 } \leqslant t \leqslant t _ { 2 }\).
4. Show that \(P\) does not return to \(O\).

2. A particle \(P\) is moving in a straight line. At time \(t\) seconds, the distance of \(P\) from a fixed point \(O\) on the line is \(x\) metres and the acceleration of \(P\) is \(( 6 - 2 t ) \mathrm { m } \mathrm { s } ^ { - 2 }\) in the direction of \(x\) increasing. When \(t = 0 , P\) is moving towards \(O\) with speed \(8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\)

1. Find the velocity of \(P\) in terms of \(t\).
2. Find the total distance travelled by \(P\) in the first 4 seconds.

6 The velocity \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) of a particle at time \(t \mathrm {~s}\) is given by \(v = t ^ { 2 } - 9\). The particle travels in a straight line and passes through a fixed point \(O\) when \(t = 2\).

1. Find the displacement of the particle from \(O\) when \(t = 0\).
2. Calculate the distance the particle travels from its position at \(t = 0\) until it changes its direction of motion.
3. Calculate the distance of the particle from \(O\) when the acceleration of the particle is \(10 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).

1. A particle, \(P\), moves along a straight line such that at time \(t\) seconds, \(t \geqslant 0\), the velocity of \(P\), \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), is modelled as

$$v = 12 + 4 t - t ^ { 2 }$$ Find

1. the magnitude of the acceleration of \(P\) when \(P\) is at instantaneous rest,
2. the distance travelled by \(P\) in the interval \(0 \leqslant t \leqslant 3\)

1. A fixed point \(O\) lies on a straight line.

A particle \(P\) moves along the straight line.
At time \(t\) seconds, \(t \geqslant 0\), the distance, \(s\) metres, of \(P\) from \(O\) is given by $$s = \frac { 1 } { 3 } t ^ { 3 } - \frac { 5 } { 2 } t ^ { 2 } + 6 t$$

1. Find the acceleration of \(P\) at each of the times when \(P\) is at instantaneous rest.
2. Find the total distance travelled by \(P\) in the interval \(0 \leqslant t \leqslant 4\)

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

A fixed point \(O\) lies on a straight line.
A particle \(P\) moves along the straight line such that at time \(t\) seconds, \(t \geqslant 0\), after passing through \(O\), the velocity of \(P , v \mathrm {~ms} ^ { - 1 }\), is modelled as $$v = 15 - t ^ { 2 } - 2 t$$

1. Verify that \(P\) comes to instantaneous rest when \(t = 3\)
2. Find the magnitude of the acceleration of \(P\) when \(t = 3\)
3. Find the total distance travelled by \(P\) in the interval \(0 \leqslant t \leqslant 4\)

1. A particle \(P\) moves along a straight line.

At time \(t\) seconds, the velocity \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) of \(P\) is modelled as $$v = 10 t - t ^ { 2 } - k \quad t \geqslant 0$$ where \(k\) is a constant.

1. Find the acceleration of \(P\) at time \(t\) seconds. The particle \(P\) is instantaneously at rest when \(t = 6\)
2. Find the other value of \(t\) when \(P\) is instantaneously at rest.
3. Find the total distance travelled by \(P\) in the interval \(0 \leqslant t \leqslant 6\)

10 A particle \(P\) is moving in a straight line. At time \(t\) seconds \(P\) has velocity \(v \mathrm {~ms} ^ { - 1 }\) where \(v = ( 2 t + 1 ) ( 3 - t )\).

1. Find the deceleration of \(P\) when \(t = 4\).
2. State the positive value of \(t\) for which \(P\) is instantaneously at rest.
3. Find the total distance that \(P\) travels between times \(t = 0\) and \(t = 4\).

10 A particle \(P\) is moving in a straight line. At time \(t\) seconds, where \(t \geqslant 0 , P\) has velocity \(v \mathrm {~ms} ^ { - 1 }\) and acceleration \(a \mathrm {~ms} ^ { - 2 }\) where \(a = 4 t - 9\). It is given that \(v = 2\) when \(t = 1\).

1. Find an expression for \(v\) in terms of \(t\). The particle \(P\) is instantaneously at rest when \(t = t _ { 1 }\) and \(t = t _ { 2 }\), where \(t _ { 1 } < t _ { 2 }\).
2. Find the values of \(t _ { 1 }\) and \(t _ { 2 }\).
3. Determine the total distance travelled by \(P\) between times \(t = 0\) and \(t = t _ { 2 }\).

7 The velocity of a particle moving in a straight line is modelled by \(\mathbf { v } = 0.6 \mathbf { t } ^ { 2 } - 2.1 \mathbf { t } + 1.5\) where \(v\) is the velocity in metres per second and \(t\) is the time in seconds.

1. Determine the times at which the particle is stationary.
2. Find the acceleration of the particle at the first of the times at which it is stationary.
3. Find the distance travelled by the particle between the times at which it is stationary.

6 In this question you must show detailed reasoning.
A particle moves in a straight line. Its velocity \(v \mathrm {~ms} ^ { - 1 }\) after \(t \mathrm {~s}\) is given by \(\mathrm { v } = \mathrm { t } ^ { 3 } - 5 \mathrm { t } ^ { 2 }\).

1. Find the times at which the particle is stationary.
2. Find the total distance travelled by the particle in the first 6 seconds.

6 Fig. 7 is a sketch of part of the velocity-time graph for the motion of an insect walking in a straight line. Its velocity, \(v \mathrm {~m} \mathrm {~s} { } ^ { 1 }\), at time \(t\) seconds for the time interval \(- 3 \leqslant t \leqslant 5\) is given by $$v = t ^ { 2 } - 2 t - 8 .$$ \begin{figure}[h] \end{figure}

1. Write down the velocity of the insect when \(t = 0\).
2. Show that the insect is instantaneously at rest when \(t = - 2\) and when \(t = 4\).
3. Determine the velocity of the insect when its acceleration is zero. Write down the coordinates of the point A shown in Fig. 7.
4. Calculate the distance travelled by the insect from \(t = 1\) to \(t = 4\).
5. Write down the distance travelled by the insect in the time interval \(- 2 \leqslant t \leqslant 4\).
6. How far does the insect walk in the time interval \(1 \leqslant t \leqslant 5\) ?

7. A particle \(P\) moves in a straight line so that its displacement \(s\) metres from a fixed point \(O\) at time \(t\) seconds is given by the formula \(s = t ^ { 3 } - 7 t ^ { 2 } + 8 t\).

1. Find the values of \(t\) when the velocity of \(P\) equals zero, and briefly describe what is happening to \(P\) at these times.
2. Find the distance travelled by \(P\) between the times \(t = 3\) and \(t = 5\).
3. Find the value of \(t\) when the acceleration of \(P\) is \(- 2 \mathrm {~ms} ^ { - 2 }\). Briefly explain the significance of a negative acceleration at this time.

3. A particle moves in a straight horizontal line such that its velocity, \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), at time \(t\) seconds is given by \(v = 2 t ^ { 2 } - 9 t + 4\). Initially, the particle has displacement 9 m from a fixed point \(O\) on the line.

1. Find the initial velocity of the particle.
2. Show that the particle is at rest when \(t = 4\) and find the other value of \(t\) when it is at rest.
3. Find the displacement of the particle from \(O\) when \(t = 6\).