Velocity from displacement differentiation

4 A particle moves in a straight line. Its displacement \(t\) seconds after leaving the fixed point \(O\) is \(x\) metres, where \(x = \frac { 1 } { 2 } t ^ { 2 } + \frac { 1 } { 30 } t ^ { 3 }\). Find

1. the speed of the particle when \(t = 10\),
2. the value of \(t\) for which the acceleration of the particle is twice its initial acceleration.

2 A particle moves in a straight line. Its displacement \(t \mathrm {~s}\) after leaving a fixed point \(O\) on the line is \(s \mathrm {~m}\), where \(s = 2 t ^ { 2 } - \frac { 80 } { 3 } t ^ { \frac { 3 } { 2 } }\).

1. Find the time at which the acceleration of the particle is zero.
2. Find the displacement and velocity of the particle at this instant.

1 A particle moves along a straight line through the origin. At time \(t\), the displacement, \(s\), of the particle from the origin is given by $$s = 5 t ^ { 2 } + 3 \cos 4 t$$ Find the velocity of the particle at time \(t\).

1 A particle, of mass 3 kg , moves along a straight line. At time \(t\) seconds, the displacement, \(s\) metres, of the particle from the origin is given by $$s = 8 t ^ { 3 } + 15$$

1. Find the velocity of the particle at time \(t\).
2. Find the magnitude of the resultant force acting on the particle when \(t = 2\).

2. A particle \(P\) moves along the \(x\)-axis such that its displacement, \(x\) metres, from the origin \(O\) at time \(t\) seconds is given by $$x = 2 + t - \frac { 1 } { 10 } \mathrm { e } ^ { t }$$

1. Find the distance of \(P\) from \(O\) when \(t = 0\).
2. Find, correct to 1 decimal place, the value of \(t\) when the velocity of \(P\) is zero.
   (4 marks)

1. A student is attempting to model the expansion of an airbag in a car following a collision.

The student considers the displacement from the steering column, \(s\) metres, of a point \(P\) on the airbag \(t\) seconds after a collision and uses the formula $$s = \mathrm { e } ^ { 3 t } - 1 , \quad 0 \leq t \leq 0.1$$ Using this model,

1. find, correct to the nearest centimetre, the maximum displacement of \(P\),
2. find the initial velocity of \(P\),
3. find the acceleration of \(P\) in terms of \(t\).
4. Explain why this model is unlikely to be realistic.

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

1. Find an expression for \(v\) in terms of \(t\). [2]
2. Find the two values of \(t\) for which \(P\) is at instantaneous rest. [2]
3. Find the minimum velocity of \(P\). [3]

A particle moves in a straight line. The displacement of the particle at time \(t\) s is \(s\) m, where $$s = t^3 - 6t^2 + 4t.$$ Find the velocity of the particle at the instant when its acceleration is zero. [4]

A particle \(P\), of mass 0.4 kg, moves in a straight line such that, at time \(t\) seconds after passing through a fixed point \(O\), its distance from \(O\) is \(x\) metres, where \(x = 3t^2 + 8t\).

1. Show that \(P\) never returns to \(O\). [2 marks]
2. Find the value of \(t\) when \(P\) has velocity 20 ms\(^{-1}\). [3 marks]
3. Show that the force acting on \(P\) is constant, and find its magnitude. [3 marks]