Variable acceleration with initial conditions

3. A particle moves along the \(x\)-axis. At time \(t = 0\) the particle passes through the origin with speed \(8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) in the positive \(x\)-direction. The acceleration of the particle at time \(t\) seconds, \(t \geqslant 0\), is \(\left( 4 t ^ { 3 } - 12 t \right) \mathrm { m } \mathrm { s } ^ { - 2 }\) in the positive \(x\)-direction. Find

1. the velocity of the particle at time \(t\) seconds,
2. the displacement of the particle from the origin at time \(t\) seconds,
3. the values of \(t\) at which the particle is instantaneously at rest.

1. A particle \(P\) moves on the \(x\)-axis. The acceleration of \(P\) at time \(t\) seconds is \(( t - 4 ) \mathrm { m } \mathrm { s } ^ { - 2 }\) in the positive \(x\)-direction. The velocity of \(P\) at time \(t\) seconds is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\). When \(t = 0 , v = 6\).

Find

1. \(v\) in terms of \(t\),
2. the values of \(t\) when \(P\) is instantaneously at rest,
3. the distance between the two points at which \(P\) is instantaneously at rest.

2. At time \(t\) seconds, where \(t \geqslant 0\), a particle \(P\) is moving on a horizontal plane with acceleration \(\left[ \left( 3 t ^ { 2 } - 4 t \right) \mathbf { i } + ( 6 t - 5 ) \mathbf { j } \right] \mathrm { m } \mathrm { s } ^ { - 2 }\). When \(t = 3\) the velocity of \(P\) is \(( 11 \mathbf { i } + 10 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\). Find

1. the velocity of \(P\) at time \(t\) seconds,
2. the speed of \(P\) when it is moving parallel to the vector \(\mathbf { i }\).

6. At time \(t\) seconds the acceleration, a \(\mathrm { m } \mathrm { s } ^ { - 2 }\), of a particle \(P\) relative to a fixed origin \(O\), is given by \(\mathbf { a } = 2 \mathbf { i } + 6 t \mathbf { j }\). Initially the velocity of \(P\) is \(( 2 \mathbf { i } - 4 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\).

1. Find the velocity of \(P\) at time \(t\) seconds. At time \(t = 2\) seconds the particle \(P\) is given an impulse ( \(3 \mathbf { i } - 1.5 \mathbf { j }\) ) Ns. Given that the particle \(P\) has mass 0.5 kg ,
2. find the speed of \(P\) immediately after the impulse has been applied.

2. A particle \(P\) is moving in a straight line. At time \(t\) seconds, the distance of \(P\) from a fixed point \(O\) on the line is \(x\) metres and the acceleration of \(P\) is \(( 6 - 2 t ) \mathrm { m } \mathrm { s } ^ { - 2 }\) in the direction of \(x\) increasing. When \(t = 0 , P\) is moving towards \(O\) with speed \(8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\)

1. Find the velocity of \(P\) in terms of \(t\).
2. Find the total distance travelled by \(P\) in the first 4 seconds.

3. At time \(t = 0\), a particle \(P\) is at the origin \(O\), moving with speed \(8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) in the positive \(x\) direction. At time \(t\) seconds, \(t \geqslant 0\), the acceleration of \(P\) has magnitude \(2 ( t + 4 ) ^ { - \frac { 1 } { 2 } } \mathrm {~ms} ^ { - 2 }\) and is directed towards \(O\).

1. Show that, at time \(t\) seconds, the velocity of \(P\) is \(16 - 4 ( t + 4 ) ^ { \frac { 1 } { 2 } } \mathrm {~ms} ^ { - 1 }\)
2. Find the distance of \(P\) from \(O\) when \(P\) comes to instantaneous rest.

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2. A particle \(P\) moves along the \(x\)-axis. At time \(t\) seconds its acceleration is \(\left( - 4 \mathrm { e } ^ { - 2 t } \right) \mathrm { m } \mathrm { s } ^ { - 2 }\) in the direction of \(x\) increasing. When \(t = 0 , P\) is at the origin \(O\) and is moving with speed \(1 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) in the direction of \(x\) increasing.

1. Find an expression for the velocity of \(P\) at time \(t\).
2. Find the distance of \(P\) from \(O\) when \(P\) comes to instantaneous rest.
   (6)

2. A particle \(P\) moves along the \(x\)-axis. At time \(t\) seconds the velocity of \(P\) is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and its acceleration is \(2 \sin \frac { 1 } { 2 } t \mathrm {~m} \mathrm {~s} ^ { - 2 }\), both measured in the direction of \(O x\). Given that \(v = 4\) when \(t = 0\),

1. find \(v\) in terms of \(t\),
2. calculate the distance travelled by \(P\) between the times \(t = 0\) and \(t = \frac { \pi } { 2 }\).

3 A car is travelling along a straight horizontal road with velocity \(32.5 \mathrm {~ms} ^ { - 1 }\). The driver applies the brakes and the car decelerates at \(( 8 - 0.6 t ) \mathrm { ms } ^ { - 2 }\), where \(t \mathrm {~s}\) is the time which has elapsed since the brakes were first applied.

1. Show that, while the car is decelerating, its velocity is \(\left( 32.5 - 8 t + 0.3 t ^ { 2 } \right) \mathrm { m } \mathrm { s } ^ { - 1 }\).
2. Find the time taken to bring the car to rest.
3. Show that the distance travelled while the car is decelerating is 75 m .

4 The acceleration of a particle \(P\) moving in a straight line is \(\left( t ^ { 2 } - 9 t + 18 \right) \mathrm { ms } ^ { - 2 }\), where \(t\) is the time in seconds.

1. Find the values of \(t\) for which the acceleration is zero.
2. It is given that when \(t = 3\) the velocity of \(P\) is \(9 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Find the velocity of \(P\) when \(t = 0\).
3. Show that the direction of motion of \(P\) changes before \(t = 1\).

2 A particle is moving along a straight line and its position is relative to an origin on the line. At time \(t \mathrm {~s}\), the particle's acceleration, \(a \mathrm {~m} \mathrm {~s} ^ { - 2 }\), is given by $$a = 6 t - 12$$ At \(t = 0\) the velocity of the particle is \(+ 9 \mathrm {~ms} ^ { - 1 }\) and its position is - 2 m .

1. Find an expression for the velocity of the particle at time \(t \mathrm {~s}\) and verify that it is stationary when \(t = 3\).
2. Find the position of the particle when \(t = 2\).

16 A particle, of mass 400 grams, is initially at rest at the point \(O\).
The particle starts to move in a straight line so that its velocity, \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), at time \(t\) seconds is given by $$v = 6 t ^ { 2 } - 12 t ^ { 3 } \text { for } t > 0$$ 16

1. Find an expression, in terms of \(t\), for the force acting on the particle.
   [0pt] [3 marks] 16
2. Find the time when the particle next passes through \(O\).
   [0pt] [5 marks] In this question use \(\boldsymbol { g } = 9.8 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).
   A van of mass 1300 kg and a crate of mass 300 kg are connected by a light inextensible rope.
   The rope passes over a light smooth pulley, as shown in the diagram.
   The rope between the pulley and the van is horizontal.
   \includegraphics[max width=\textwidth, alt={}, center]{3176ee0c-fba2-4878-af3a-c3ac092bbc1f-20_515_766_685_607} Initially, the van is at rest and the crate rests on the lower level. The rope is taut.
   The van moves away from the pulley to lift the crate from the lower level.
   The van's engine produces a constant driving force of 5000 N .
   A constant resistance force of magnitude 780 N acts on the van.
   Assume there is no resistance force acting on the crate.

10 A particle \(P\) is moving in a straight line. At time \(t\) seconds, where \(t \geqslant 0 , P\) has velocity \(v \mathrm {~ms} ^ { - 1 }\) and acceleration \(a \mathrm {~ms} ^ { - 2 }\) where \(a = 4 t - 9\). It is given that \(v = 2\) when \(t = 1\).

1. Find an expression for \(v\) in terms of \(t\). The particle \(P\) is instantaneously at rest when \(t = t _ { 1 }\) and \(t = t _ { 2 }\), where \(t _ { 1 } < t _ { 2 }\).
2. Find the values of \(t _ { 1 }\) and \(t _ { 2 }\).
3. Determine the total distance travelled by \(P\) between times \(t = 0\) and \(t = t _ { 2 }\).

3 Two girls, Marie and Nina, are members of an Olympic hockey team. They are doing fitness training.
Marie runs along a straight line at a constant speed of \(6 \mathrm {~ms} ^ { - 1 }\).
Nina is stationary at a point O on the line until Marie passes her. Nina immediately runs after Marie until she catches up with her. The time, \(t \mathrm {~s}\), is measured from the moment when Nina starts running. So when \(t = 0\), both girls are at O .
Nina's acceleration, \(a \mathrm {~ms} ^ { - 2 }\), is given by $$\begin{array} { l l } a = 4 - t & \text { for } 0 \leqslant t \leqslant 4 , \\ a = 0 & \text { for } t > 4 . \end{array}$$

1. Show that Nina's speed, \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), is given by $$\begin{array} { l l } v = 4 t - \frac { 1 } { 2 } t ^ { 2 } & \text { for } 0 \leqslant t \leqslant 4 , \\ v = 8 & \text { for } t > 4 . \end{array}$$
2. Find an expression for the distance Nina has run at time \(t\), for \(0 \leqslant t \leqslant 4\). Find how far Nina has run when \(t = 4\) and when \(t = 5 \frac { 1 } { 3 }\).
3. Show that Nina catches up with Marie when \(t = 5 \frac { 1 } { 3 }\).

3 A particle is moving along a straight line and its position is relative to an origin on the line. At time \(t \mathrm {~s}\), the particle's acceleration, \(a \mathrm {~m} \mathrm {~s} ^ { - 2 }\), is given by $$a = 6 t - 12 .$$ At \(t = 0\) the velocity of the particle is \(+ 9 \mathrm {~ms} ^ { - 1 }\) and its position is - 2 m .

1. Find an expression for the velocity of the particle at time \(t \mathrm {~s}\) and verify that it is stationary when \(t = 3\).
2. Find the position of the particle when \(t = 2\).

2 The acceleration of a particle of mass 4 kg is given by \(\mathbf { a } = ( 9 \mathbf { i } - 4 t \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { 2 }\), where \(\mathbf { i }\) and \(\mathbf { j }\) are unit vectors and \(t\) is the time in seconds.

1. Find the acceleration of the particle when \(t = 0\) and also when \(t = 3\).
2. Calculate the force acting on the particle when \(t = 3\). The particle has velocity \(( 4 \mathbf { i } + 2 \mathbf { j } ) \mathrm { m } \mathrm { s } { } ^ { 1 }\) when \(t = 1\).
3. Find an expression for the velocity of the particle at time \(t\).

4. A particle \(P\) moves in a straight horizontal line such that its acceleration at time \(t\) seconds is proportional to \(\left( 3 t ^ { 2 } - 5 \right)\). Given that at time \(t = 0 , P\) is at rest at the origin \(O\) and that at time \(t = 3\), its velocity is \(3 \mathrm {~ms} ^ { - 1 }\),

1. find, in \(\mathrm { m } \mathrm { s } ^ { - 2 }\), the acceleration of \(P\) in terms of \(t\),
2. show that the displacement of the particle, \(s\) metres, from \(O\) at time \(t\) is given by $$s = \frac { 1 } { 16 } t ^ { 2 } \left( t ^ { 2 } - 10 \right)$$ (4 marks)

3. At time \(t\) seconds the acceleration, \(a \mathrm {~ms} ^ { - 2 }\), of a particle is given by $$a = \frac { 4 } { ( 1 + t ) ^ { 3 } }$$ When \(t = 0\), the particle has velocity \(1 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and displacement 3 m from a fixed origin \(O\).

1. Find an expression for the velocity of the particle in terms of \(t\).
2. Show that when \(t = 3\) the particle is 10.5 m from \(O\).

1. At time \(t = 0\), a particle \(P\) of mass 3 kg is at rest at the point \(A\) with position vector \(( \mathbf { j } - 3 \mathbf { k } ) \mathrm { m }\). Two constant forces \(\mathbf { F } _ { 1 }\) and \(\mathbf { F } _ { 2 }\) then act on the particle \(P\) and it passes through the point \(B\) with position vector \(( 8 \mathbf { i } - 3 \mathbf { j } + 5 \mathbf { k } ) \mathrm { m }\).

Given that \(\mathbf { F } _ { 1 } = ( 4 \mathbf { i } - 2 \mathbf { j } + 5 \mathbf { k } ) \mathrm { N }\) and \(\mathbf { F } _ { 2 } = ( 8 \mathbf { i } - 4 \mathbf { j } + 7 \mathbf { k } ) \mathrm { N }\) and that \(\mathbf { F } _ { 1 }\) and \(\mathbf { F } _ { 2 }\) are the only two forces acting on \(P\), find the velocity of \(P\) as it passes through \(B\), giving your answer as a vector.

1. A particle, \(P\), moves on the \(x\)-axis. At time \(t\) seconds, \(t \geqslant 0\), the velocity of \(P\) is \(v \mathrm {~ms} ^ { - 1 }\) in the direction of \(x\) increasing and the acceleration of \(P\) is \(a \mathrm {~m} \mathrm {~s} ^ { - 2 }\) in the direction of \(x\) increasing.

When \(t = 0\) the particle is at rest at the origin \(O\).
Given that \(a = \frac { 5 } { 2 } ( 5 - v )\)

1. show that \(v = 5 \left( 1 - \mathrm { e } ^ { - 2.5 t } \right)\)
2. state the limiting value of \(v\) as \(t\) increases. At the instant when \(v = 2.5\), the particle is \(d\) metres from \(O\).
3. Show that \(d = 2 \ln 2 - 1\)

1. A car moves in a straight line along a horizontal road. The car is modelled as a particle. At time \(t\) seconds, where \(t \geqslant 0\), the speed of the car is \(v \mathrm {~ms} ^ { - 1 }\)

At the instant when \(t = 0\), the car passes through the point \(A\) with speed \(2 \mathrm {~ms} ^ { - 1 }\)
The acceleration, \(a \mathrm {~m} \mathrm {~s} ^ { - 2 }\), of the car is modelled by $$a = \frac { 4 } { 2 + v }$$ in the direction of motion of the car.

1. Use algebraic integration to show that \(v = \sqrt { 8 t + 16 } - 2\) At the instant when the car passes through the point \(B\), the speed of the car is \(4 \mathrm {~ms} ^ { - 1 }\)
2. Use algebraic integration to find the distance \(A B\).

1. At time \(t = 0\), a toy electric car is at rest at a fixed point \(O\). The car then moves in a horizontal straight line so that at time \(t\) seconds \(( t > 0 )\) after leaving \(O\), the velocity of the car is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and the acceleration of the car is modelled as \(( p + q v ) \mathrm { ms } ^ { - 2 }\), where \(p\) and \(q\) are constants.

When \(t = 0\), the acceleration of the car is \(3 \mathrm {~ms} ^ { - 2 }\)
When \(t = T\), the acceleration of the car is \(\frac { 1 } { 2 } \mathrm {~ms} ^ { - 2 }\) and \(v = 4\)

1. Show that $$8 \frac { \mathrm {~d} v } { \mathrm {~d} t } = ( 24 - 5 v )$$
2. Find the exact value of \(T\), simplifying your answer.

1. A particle \(P\) is moving along the \(x\)-axis. At time \(t\) seconds, \(P\) has velocity \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) in the positive \(x\) direction and acceleration \(a \mathrm {~ms} ^ { - 2 }\) in the positive \(x\) direction.

In a model of the motion of \(P\) $$a = 4 - 3 v$$ When \(t = 0 , v = 0\)

1. Use integration to show that \(v = k \left( 1 - \mathrm { e } ^ { - 3 t } \right)\), where \(k\) is a constant to be found. When \(t = 0 , P\) is at the origin \(O\)
2. Find, in terms of \(t\) only, the distance of \(P\) from \(O\) at time \(t\) seconds.

1. In this question you must show all stages of your working. Solutions relying entirely on calculator technology are not acceptable.

A particle \(P\) moves along a straight line. Initially \(P\) is at rest at the point \(O\) on the line. At time \(t\) seconds, where \(t \geqslant 0\)

• the displacement of \(P\) from \(O\) is \(x\) metres
• the velocity of \(P\) is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) in the positive \(x\) direction
• the acceleration of \(P\) is \(\frac { 96 } { ( 3 t + 5 ) ^ { 3 } } \mathrm {~ms} ^ { - 2 }\) in the positive \(x\) direction
  1. Show that, at time \(t\) seconds, \(v = p - \frac { q } { ( 3 t + 5 ) ^ { 2 } }\), where \(p\) and \(q\) are constants to be determined.
  2. Find the limiting value of \(v\) as \(t\) increases.
  3. Find the value of \(x\) when \(t = 2\)