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

1 A particle moves in a horizontal plane, in which the unit vectors \(\mathbf { i }\) and \(\mathbf { j }\) are directed east and north respectively. At time \(t\) seconds, its position vector, \(\mathbf { r }\) metres, is given by $$\mathbf { r } = \left( 2 t ^ { 3 } - t ^ { 2 } + 6 \right) \mathbf { i } + \left( 8 - 4 t ^ { 3 } + t \right) \mathbf { j }$$

1. Find an expression for the velocity of the particle at time \(t\).
2. 1. Find the velocity of the particle when \(t = \frac { 1 } { 3 }\).
   2. State the direction in which the particle is travelling at this time.
3. Find the acceleration of the particle when \(t = 4\).
4. The mass of the particle is 6 kg . Find the magnitude of the resultant force on the particle when \(t = 4\).

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 \(t = 2\).

1. A particle \(P\) of mass 2 kg is subjected to a force \(\mathbf { F }\) such that its displacement, \(\mathbf { r }\) metres, from a fixed origin, \(O\), at time \(t\) seconds is given by

$$\mathbf { r } = \left( 3 t ^ { 2 } - 4 \right) \mathbf { i } + \left( 3 - 4 t ^ { 2 } \right) \mathbf { j }$$

1. Show that the acceleration of \(P\) is constant.
2. Find the magnitude of \(\mathbf { F }\).

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.

5 A particle P of mass 4 kilograms moves in such a way that its position vector at time \(t\) seconds is r metres, where
\(\mathbf { r } = 3 t \mathbf { i } + 2 \mathrm { e } ^ { - 3 t } \mathbf { j }\).

1. Find the initial kinetic energy of P .
2. Find the time when the acceleration of P is 2 metres per second squared. Section B (93 marks)

1 In this question, \(\mathbf { i }\) and \(\mathbf { j }\) represent unit vectors due east and due north respectively. Sarah is going for a walk. She leaves her house and walks directly to the shop. She then walks directly from the shop to the park. Relative to her house:

• the shop has position vector \(\left( - \frac { 2 } { 3 } \mathbf { j } \right) \mathrm { km }\),
• the park is 2 km away on a bearing of \(060 ^ { \circ }\).
  a) Show that the position vector of the park relative to the house is \(( \sqrt { 3 } \mathbf { i } + \mathbf { j } ) \mathrm { km }\).
  b) Determine the total distance walked by Sarah from her house to the park.
  c) By considering a modelling assumption you have made, explain why the answer you found in part (b) may not be the actual distance that Sarah walked.

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.

10 A student is attempting to model the flight of a boomerang.
She throws the boomerang from a fixed point \(O\) and catches it when it returns to \(O\).
She suggests the model for the displacement, \(s\) metres, after \(t\) seconds in given by
\(s = 9 t ^ { 2 } - \frac { 3 } { 2 } t ^ { 3 } , 0 \leq t \leq 6\). For this model,

1. determine what happens at \(t = 6\),
2. find the greatest displacement of the boomerang from \(O\),
3. find the velocity of the boomerang 1 second before the student catches it,
4. find the acceleration of the boomerang 1 second before the student catches it.

3 A particle moves on a horizontal plane, in which the unit vectors \(\mathbf { i }\) and \(\mathbf { j }\) are directed east and north respectively. At time \(t\) seconds, the position vector of the particle is \(\mathbf { r }\) metres, where $$\mathbf { r } = \left( 2 \mathrm { e } ^ { \frac { 1 } { 2 } t } - 8 t + 5 \right) \mathbf { i } + \left( t ^ { 2 } - 6 t \right) \mathbf { j }$$

1. Find an expression for the velocity of the particle at time \(t\).
2. 1. Find the speed of the particle when \(t = 3\).
   2. State the direction in which the particle is travelling when \(t = 3\).
3. Find the acceleration of the particle when \(t = 3\).
4. The mass of the particle is 7 kg . Find the magnitude of the resultant force on the particle when \(t = 3\).

11 In this question the unit vectors \(\mathbf { i }\) and \(\mathbf { j }\) are in the directions east and north respectively. A particle of mass 0.12 kg is moving so that its position vector \(\mathbf { r }\) metres at time \(t\) seconds is given by \(\mathbf { r } = 2 t ^ { 3 } \mathbf { i } + \left( 5 t ^ { 2 } - 4 t \right) \mathbf { j }\).

1. Show that when \(t = 0.7\) the bearing on which the particle is moving is approximately \(044 ^ { \circ }\).
2. Find the magnitude of the resultant force acting on the particle at the instant when \(t = 0.7\).
3. Determine the times at which the particle is moving on a bearing of \(045 ^ { \circ }\).

14 At time \(t\) seconds a particle, \(P\), has position vector \(\mathbf { r }\) metres, with respect to a fixed origin, such that $$\mathbf { r } = \left( t ^ { 3 } - 5 t ^ { 2 } \right) \mathbf { i } + \left( 8 t - t ^ { 2 } \right) \mathbf { j }$$ 14

1. Find the exact speed of \(P\) when \(t = 2\)
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2. Bella claims that the magnitude of acceleration of \(P\) will never be zero.
   Determine whether Bella's claim is correct.
   Fully justify your answer.

17 A particle is moving such that its position vector, \(\mathbf { r }\) metres, at time \(t\) seconds, is given by $$\mathbf { r } = \mathrm { e } ^ { t } \cos t \mathbf { i } + \mathrm { e } ^ { t } \sin t \mathbf { j }$$ Show that the magnitude of the acceleration of the particle, \(a \mathrm {~ms} ^ { - 2 }\), is given by $$a = 2 \mathrm { e } ^ { t }$$ Fully justify your answer.
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1 A particle P has position vector \(\mathbf { r } \mathrm { m }\) at time \(t\) s given by \(\mathbf { r } = \left( t ^ { 3 } - 3 t ^ { 2 } \right) \mathbf { i } - \left( 4 t ^ { 2 } + 1 \right) \mathbf { j }\) for \(t \geq 0\).
Find the magnitude of the acceleration of P when \(t = 2\).