Total distance with direction changes

5. A particle \(P\) moves in a straight line with an acceleration of \(( 6 t - 10 ) \mathrm { m } \mathrm { s } ^ { - 2 }\) at time \(t\) seconds. Initially \(P\) is at \(O\), a fixed point on the line, and has velocity \(3 \mathrm {~ms} ^ { - 1 }\).

1. Find the values of \(t\) for which the velocity of \(P\) is zero.
2. Show that, during the first two seconds, \(P\) travels a distance of \(6 \frac { 26 } { 27 } \mathrm {~m}\).

1. 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

$$v = ( t - 2 ) ( 3 t - 10 ) , \quad t \geqslant 0$$ When \(t = 0 , P\) is at the origin \(O\).

1. Find the acceleration of \(P\) at time \(t\) seconds.
2. Find the total distance travelled by \(P\) in the first 2 seconds of its motion.
3. Show that \(P\) never returns to \(O\), explaining your reasoning.

A particle moves in a straight line starting from a point \(O\). The velocity \(v\) m s\(^{-1}\) of the particle \(t\) s after leaving \(O\) is given by $$v = t^3 - \frac{9}{2}t^2 + 1 \text{ for } 0 \leqslant t \leqslant 4.$$ You may assume that the velocity of the particle is positive for \(t < \frac{1}{2}\), is zero at \(t = \frac{1}{2}\) and is negative for \(t > \frac{1}{2}\).

1. Find the distance travelled between \(t = 0\) and \(t = \frac{1}{2}\). [4]
2. Find the positive value of \(t\) at which the acceleration is zero. Hence find the total distance travelled between \(t = 0\) and this instant. [4]

A particle \(P\) moves in a straight line, passing through a point \(O\) with velocity \(4.2 \text{ ms}^{-1}\). At time \(t\) s after \(P\) passes \(O\), the acceleration, \(a \text{ ms}^{-2}\), of \(P\) is given by \(a = 0.6t - 2.7\). Find the distance \(P\) travels between the times at which it is at instantaneous rest. [7]

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

1. Find the two positive values of \(t\) for which the particle is instantaneously at rest. [2]
2. Find the time at which the acceleration of the particle is greatest. [3]
3. Find the distance travelled by the particle while its velocity is positive. [4]

A particle moves in a straight line. Its displacement from a fixed point \(O\) at time \(t\) seconds is \(s\) metres, where \(s = t^3 - 9t^2 + 24t\).

1. Find expressions for the velocity \(v\) and acceleration \(a\) of the particle at time \(t\).
2. Find the values of \(t\) for which the particle is at rest.
3. Find the total distance travelled by the particle in the first \(6\) seconds.

[8]

A particle \(P\) moves on the \(x\)-axis. At time \(t\) seconds the velocity of \(P\) is \(v\) m s\(^{-1}\) in the direction of \(x\) increasing, where $$v = (t - 2)(3t - 10), \quad t \geq 0$$ When \(t = 0\), \(P\) is at the origin \(O\).

1. Find the acceleration of \(P\) when \(t = 3\) [3]
2. Find the total distance travelled by \(P\) in the first 3 seconds of its motion. [6]
3. Show that \(P\) never returns to \(O\). [2]

A particle \(P\) moves along a straight line in such a way that at time \(t\) seconds its velocity \(v\) m s\(^{-1}\) is given by $$v = \frac{1}{2}t^2 - 3t + 4$$ Find

1. the times when \(P\) is at rest, [4]
2. the total distance travelled by \(P\) between \(t = 0\) and \(t = 4\). [5]

The velocity, \(v\) ms\(^{-1}\), of a particle at time \(t\) s is given by \(v = 4t^2 - 9\).

1. Find the acceleration of the particle when it is instantaneously at rest. [3 marks]
2. Find the distance travelled by the particle from time \(t = 0\) until it comes to rest. [4 marks]