Finding when particle at rest

6 A particle moves in a straight line and passes through the point \(A\) at time \(t = 0\). The velocity of the particle at time \(t \mathrm {~s}\) after leaving \(A\) is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), where $$v = 2 t ^ { 2 } - 5 t + 3$$

1. Find the times at which the particle is instantaneously at rest. Hence or otherwise find the minimum velocity of the particle.
2. Sketch the velocity-time graph for the first 3 seconds of motion.
3. Find the distance travelled between the two times when the particle is instantaneously at rest.

3 A particle moves in a straight line starting from rest. The displacement \(s m\) of the particle from a fixed point \(O\) on the line at time \(t \mathrm {~s}\) is given by $$s = t ^ { \frac { 5 } { 2 } } - \frac { 15 } { 4 } t ^ { \frac { 3 } { 2 } } + 6$$ Find the value of \(s\) when the particle is again at rest. \includegraphics[max width=\textwidth, alt={}, center]{f9e3d562-ae3c-49cc-bc92-56956d939252-06_730_1545_280_294} The velocity of a particle at time \(t \mathrm {~s}\) after leaving a fixed point \(O\) is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The diagram shows a velocity-time graph which models the motion of the particle. The graph consists of 5 straight line segments. The particle accelerates to a speed of \(0.9 \mathrm {~ms} ^ { - 1 }\) in a period of 3 s , then travels at constant speed for 6 s , and then comes instantaneously to rest 1 s later. The particle then moves back and returns to rest at \(O\) at time \(T \mathrm {~s}\).

1. Find the distance travelled by the particle in the first 10 s of its motion.
2. Given that \(T = 12\), find the minimum velocity of the particle.
3. Given instead that the greatest speed of the particle is \(3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), find the value of \(T\) and hence find the average speed of the particle for the whole of the motion. \includegraphics[max width=\textwidth, alt={}, center]{f9e3d562-ae3c-49cc-bc92-56956d939252-08_858_563_264_794} Four coplanar forces act at a point. The magnitudes of the forces are \(F \mathrm {~N} , 10 \mathrm {~N} , 50 \mathrm {~N}\) and 40 N . The directions of the forces are as shown in the diagram.

2 A particle \(P\) moves in a straight line. At time \(t\) s after leaving a point \(O\) on the line, \(P\) has velocity \(v \mathrm {~ms} ^ { - 1 }\), where \(\mathrm { v } = 44 \mathrm { t } - 6 \mathrm { t } ^ { 2 } - 36\).

1. Find the set of values of \(t\) for which the acceleration of the particle is positive.
2. Find the two values of \(t\) at which \(P\) returns to \(O\). \includegraphics[max width=\textwidth, alt={}, center]{3eaf3652-ff91-4bae-9f20-83487d635612-04_714_796_248_635} Four coplanar forces of magnitude \(P \mathrm {~N} , 10 \mathrm {~N} , 16 \mathrm {~N}\) and 2 N act at a point in the directions shown in the diagram. It is given that the forces are in equilibrium. Find the values of \(\theta\) and \(P\).

5 A particle \(P\) moves in a straight line. It starts at a point \(O\) on the line and at time \(t\) s after leaving \(O\) it has velocity \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), where \(v = 4 t ^ { 2 } - 20 t + 21\).

1. Find the values of \(t\) for which \(P\) is at instantaneous rest.
2. Find the initial acceleration of \(P\).
3. Find the minimum velocity of \(P\).
4. Find the distance travelled by \(P\) during the time when its velocity is negative.

6 A particle travels in a straight line from a point \(P\) to a point \(Q\). Its velocity \(t\) seconds after leaving \(P\) is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), where \(v = 4 t - \frac { 1 } { 16 } t ^ { 3 }\). The distance \(P Q\) is 64 m .

1. Find the time taken for the particle to travel from \(P\) to \(Q\).
2. Find the set of values of \(t\) for which the acceleration of the particle is positive.

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.

6 A particle \(P\) moves in a straight line. It starts at a point \(O\) on the line and at time \(t\) s after leaving \(O\) it has a velocity \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), where \(v = 6 t ^ { 2 } - 30 t + 24\).

1. Find the set of values of \(t\) for which the acceleration of the particle is negative.
2. Find the distance between the two positions at which \(P\) is at instantaneous rest.
3. Find the two positive values of \(t\) at which \(P\) passes through \(O\).

7 A particle starts from rest and moves in a straight line. The velocity of the particle at time \(t \mathrm {~s}\) after the start is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), where $$v = - 0.01 t ^ { 3 } + 0.22 t ^ { 2 } - 0.4 t$$

1. Find the two positive values of \(t\) for which the particle is instantaneously at rest.
2. Find the time at which the acceleration of the particle is greatest.
3. Find the distance travelled by the particle while its velocity is positive.

4 A particle \(P\) moving in a straight line has velocity \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at time \(t \mathrm {~s}\) after passing through a fixed point \(O\). It is given that \(v = 3.2 - 0.2 t ^ { 2 }\) for \(0 \leqslant t \leqslant 5\). Calculate

1. the value of \(t\) when \(P\) is at instantaneous rest,
2. the acceleration of \(P\) when it is at instantaneous rest,
3. the greatest distance of \(P\) from \(O\).

2 A particle moves along a straight line containing a point O . Its displacement, \(x \mathrm {~m}\), from O at time \(t\) seconds is given by $$x = 12 t - t ^ { 3 } , \text { where } - 10 \leqslant t \leqslant 10$$ Find the values of \(x\) for which the velocity of the particle is zero.

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 \(t _ { 2 }\).

1. A particle \(P\) moves along a straight line. The fixed point \(O\) is on the line. At time \(t\) seconds, \(t > 0\), the displacement of \(P\) from \(O\) is \(x\) metres, where

$$x = 2 t ^ { 3 } - 21 t ^ { 2 } + 60 t$$ Find

1. the values of \(t\) for which \(P\) is instantaneously at rest
2. the distance travelled by \(P\) in the interval \(1 \leqslant t \leqslant 3\)
3. the magnitude of the acceleration of \(P\) at the instant when \(t = 3\)

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 \(O\).

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

$$v = ( 7 - 2 t ) ( t + 2 )$$

1. Find the value of \(t\) at the instant when \(P\) stops accelerating.
2. Find the distance of \(P\) from \(O\) at the instant when \(P\) changes its direction of motion. In this question, solutions relying on calculator technology are not acceptable.

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. 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 \(0 \leqslant t \leqslant 6\)

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 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.

5 Particle P moves on a straight line that contains the point O .
At time \(t\) seconds the displacement of P from O is \(s\) metres, where \(s = t ^ { 3 } - 3 t ^ { 2 } + 3\).

1. Determine the times when the particle has zero velocity.
2. Find the distances of P from O at the times when it has zero velocity.

4 A particle moves along a straight line through an origin O . Initially the particle is at O .
At time \(t \mathrm {~s}\), its displacement from O is \(x \mathrm {~m}\) and its velocity, \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), is given by $$v = 24 - 18 t + 3 t ^ { 2 }$$

1. Find the times, \(T _ { 1 } \mathrm {~s}\) and \(T _ { 2 } \mathrm {~s}\) (where \(T _ { 2 } > T _ { 1 }\) ), at which the particle is stationary.
2. Find an expression for \(x\) at time \(t\) s. Show that when \(t = T _ { 1 } , x = 20\) and find the value of \(x\) when \(t = T _ { 2 }\).

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.