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.

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.

2 A motorcyclist starts from rest at \(A\) and travels in a straight line until he comes to rest again at \(B\). The velocity of the motorcyclist \(t\) seconds after leaving \(A\) is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), where \(v = t - 0.01 t ^ { 2 }\). Find

1. the time taken for the motorcyclist to travel from \(A\) to \(B\),
2. the distance \(A B\).

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.

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 \(O\).

6 A particle \(P\) starts from rest at a point \(O\) of a straight line and moves along the line. The displacement of the particle at time \(t \mathrm {~s}\) after leaving \(O\) is \(x \mathrm {~m}\), where $$x = 0.08 t ^ { 2 } - 0.0002 t ^ { 3 }$$

1. Find the value of \(t\) when \(P\) returns to \(O\) and find the speed of \(P\) as it passes through \(O\) on its return.
2. For the motion of \(P\) until the instant it returns to \(O\), find
   1. the total distance travelled,
   2. the average speed.

5 A particle \(P\) starts from a fixed point \(O\) and moves in a straight line. At time \(t\) s after leaving \(O\), the velocity \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) of \(P\) is given by \(v = 6 t - 0.3 t ^ { 2 }\). The particle comes to instantaneous rest at point \(X\).

1. Find the distance \(O X\). A second particle \(Q\) starts from rest from \(O\), at the same instant as \(P\), and also travels in a straight line. The acceleration \(a \mathrm {~m} \mathrm {~s} ^ { - 2 }\) of \(Q\) is given by \(a = k - 12 t\), where \(k\) is a constant. The displacement of \(Q\) from \(O\) is 400 m when \(t = 10\).
2. Find the value of \(k\).

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.

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 \(t = 3\)

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.

3. A particle moves along a straight horizontal track such that its displacement, \(s\) metres, from a fixed point \(O\) on the line after \(t\) seconds is given by $$s = 2 t ^ { 3 } - 13 t ^ { 2 } + 20 t$$

1. Find the values of \(t\) for which the particle is at \(O\).
2. Find the values of \(t\) at which the particle comes instantaneously to rest.

1. A bird leaves its nest at time \(t = 0\) for a short flight along a straight line.

The bird then returns to its nest.
The bird is modelled as a particle moving in a straight horizontal line.
The distance, \(s\) metres, of the bird from its nest at time \(t\) seconds is given by $$s = \frac { 1 } { 10 } \left( t ^ { 4 } - 20 t ^ { 3 } + 100 t ^ { 2 } \right) , \quad \text { where } 0 \leqslant t \leqslant 10$$

1. Explain the restriction, \(0 \leqslant t \leqslant 10\)
2. Find the distance of the bird from the nest when the bird first comes to instantaneous rest.

7 A particle travels along a straight line. Its velocity \(v \mathrm {~ms} ^ { - 1 }\) after \(t\) seconds is given by $$v = t ^ { 3 } - 9 t ^ { 2 } + 20 t$$ When \(t = 0\), the particle is at rest at \(P\).

1. Find the times, other than \(t = 0\), at which the particle is at rest.
2. Find the displacement of the particle from \(P\) when \(t = 2\).

A particle \(P\) moves along the \(x\)-axis so that its velocity \(v \mathrm {~ms} ^ { - 1 }\) at time \(t\) seconds \(( t \geqslant 0 )\) is given by $$v = 3 t ^ { 2 } - 24 t + 36$$ a) Find the values of \(t\) when \(P\) is instantaneously at rest.
b) Calculate the total distance travelled by the particle \(P\) whilst its velocity is decreasing.

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 \(t\) s is given by $$s = t^2 - \frac{15}{4}t^2 + 6.$$ Find the value of \(s\) when the particle is again at rest. [4]

At time \(t = 0\) a ball is projected vertically upwards from a point \(O\) and rises to a maximum height of 40 m above \(O\). The ball is modelled as a particle moving freely under gravity.

1. Show that the speed of projection is 28 m s\(^{-1}\). [3]
2. Find the times, in seconds, when the ball is 33.6 m above \(O\). [5]

A particle \(P\) moves in a straight line. At time \(t\) s the displacement \(s\) cm from a fixed point \(O\) is given by: $$s = \frac{1}{6}\left(8t^3 - 105t^2 + 144t + 540\right).$$ Find the distance between the points at which the particle is instantaneously at rest. [7]

A particle travels along a straight line. Its velocity \(v\) m s\(^{-1}\) after \(t\) seconds is given by $$v = t^3 - 9t^2 + 20t$$ When \(t = 0\), the particle is at rest at \(P\).

1. Find the times, other than \(t = 0\), at which the particle is at rest. [2]
2. Find the displacement of the particle from \(P\) when \(t = 2\). [4]

A particle travels along a straight line. Its velocity \(v\) ms\(^{-1}\) after \(t\) seconds is given by $$v = t^3 - 9t^2 + 20t$$ When \(t = 0\), the particle is at rest at \(P\).

1. Find the times, other than \(t = 0\), at which the particle is at rest. [2]
2. Find the displacement of the particle from \(P\) when \(t = 2\). [4]