Force depends on time t

4 A particle \(P\) of mass 0.25 kg moves in a straight line on a smooth horizontal surface. At time \(t \mathrm {~s}\) the velocity of \(P\) is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\). A variable force of magnitude \(3 t \mathrm {~N}\) opposes the motion of \(P\).

1. Given that \(P\) comes to rest when \(t = 3\), find \(v\) when \(t = 0\).
2. Calculate the distance travelled by \(P\) in the interval \(0 \leqslant t \leqslant 3\).

5 A particle \(P\) of mass 0.4 kg is placed at rest at a point \(A\) on a rough horizontal surface. A horizontal force, directed away from \(A\) and with magnitude \(0.6 t \mathrm {~N}\), acts on \(P\), where \(t \mathrm {~s}\) is the time after \(P\) is placed at \(A\). The coefficient of friction between \(P\) and the surface is 0.3 , and \(P\) has displacement from \(A\) of \(x \mathrm {~m}\) at time \(t \mathrm {~s}\).

1. Show that \(P\) starts to move when \(t = 2\). Show also that when \(P\) is in motion it has acceleration \(( 1.5 t - 3 ) \mathrm { m } \mathrm { s } ^ { - 2 }\).
2. Express the velocity of \(P\) in terms of \(t\), for \(t \geqslant 2\).
3. Express \(x\) in terms of \(t\), for \(t \geqslant 2\).

2. A particle \(P\) of mass 0.6 kg is moving along the positive \(x\)-axis in the positive direction. The only force acting on \(P\) acts in the direction of \(x\) increasing and has magnitude \(\left( 3 t + \frac { 1 } { 2 } \right) \mathrm { N }\), where \(t\) seconds is the time after \(P\) leaves the origin \(O\). When \(t = 0 , P\) is at rest at \(O\).

1. Find an expression, in terms of \(t\), for the velocity of \(P\) at time \(t\) seconds. The particle passes through the point \(A\) with speed \(\frac { 10 } { 3 } \mathrm {~ms} ^ { - 1 }\).
2. Find the distance \(O A\).

5. A car of mass 800 kg moves along a horizontal straight road. At time \(t\) seconds, the resultant force acting on the car has magnitude \(\frac { 48000 } { ( t + 2 ) ^ { 2 } }\) newtons, acting in the direction of the motion of the car. When \(t = 0\), the car is at rest.

1. Show that the speed of the car approaches a limiting value as \(t\) increases and find this value.
2. Find the distance moved by the car in the first 6 seconds of its motion.

1. A particle \(P\) of mass 3 kg is moving in a straight line. At time \(t\) seconds, \(0 \leqslant t \leqslant 4\), the only force acting on \(P\) is a resistance to motion of magnitude \(\left( 9 + \frac { 15 } { ( t + 1 ) ^ { 2 } } \right) \mathrm { N }\). At time \(t\) seconds the velocity of \(P\) is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\). When \(t = 4 , v = 0\).

Find the value of \(v\) when \(t = 0\).

1. A particle \(P\) of mass 0.5 kg is moving along the positive \(x\)-axis. At time \(t\) seconds, \(P\) is moving under the action of a single force of magnitude \([ 4 + \cos ( \pi t ) ] \mathrm { N }\), directed away from the origin. When \(t = 1\), the particle \(P\) is moving away from the origin with speed \(6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

Find the speed of \(P\) when \(t = 1.5\), giving your answer to 3 significant figures.

2. A particle \(P\) of mass 0.5 kg is moving along the positive \(x\)-axis in the positive \(x\)-direction. The only force on \(P\) is a force of magnitude \(\left( 2 t + \frac { 1 } { 2 } \right) \mathrm { N }\) acting in the direction of \(x\) increasing, where \(t\) seconds is the time after \(P\) leaves the origin \(O\). When \(t = 0\), \(P\) is at rest at \(O\).

1. Find an expression, in terms of \(t\), for the velocity of \(P\) at time \(t\) seconds. The particle passes through the point \(A\) with speed \(6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
2. Find the distance \(O A\).

5 A particle \(P\) of mass 3 kg moves on the \(x\)-axis under the action of a single force acting in the positive \(x\)-direction. At time \(t \mathrm {~s}\), where \(t \geqslant 0\), the displacement of \(P\) is \(x \mathrm {~m}\) and its velocity is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The magnitude of the force acting is inversely proportional to \(( t + 1 ) ^ { 2 }\). Initially \(P\) is at rest at the point where \(x = 1\). When \(t = 1 , v = 2\).

1. Show that \(\frac { \mathrm { dv } } { \mathrm { dt } } = \frac { \mathrm { k } } { 3 ( \mathrm { t } + 1 ) ^ { 2 } }\) where \(k\) is a constant.
2. Find an expression for \(v\) in terms of \(t\).
3. Find an expression for \(x\) in terms of \(t\). As \(t\) increases, \(v\) approaches a limiting value, \(\mathrm { V } _ { \mathrm { T } }\).
4. Determine how far \(P\) is from its initial position at the instant when \(v\) is \(95 \%\) of \(\mathrm { V } _ { \mathrm { T } }\).

5 The graph shows a model for the resultant horizontal force on a car, which varies as it accelerates from rest for 20 seconds. The mass of the car is 1200 kg . \includegraphics[max width=\textwidth, alt={}, center]{c02cf013-365b-44e2-8c16-aa8209cbe250-4_373_1203_445_390}

1. The acceleration of the car at time \(t\) seconds is \(a \mathrm {~m} \mathrm {~s} ^ { - 2 }\). Show that $$a = \frac { 2 } { 3 } + \frac { t } { 20 } , \text { for } 0 \leqslant t \leqslant 20$$
2. Find an expression for the velocity of the car at time \(t\).
3. Find the distance travelled by the car in the 20 seconds.
4. An alternative model assumes that the resultant force increases uniformly from 900 to 2100 newtons during the 20 seconds. Which term in your expression for the velocity would change as a result of this modification? Explain why.

4 A particle has mass 200 kg and moves on a smooth horizontal plane. A single horizontal force, \(\left( 400 \cos \left( \frac { \pi } { 2 } t \right) \mathbf { i } + 600 t ^ { 2 } \mathbf { j } \right)\) newtons, acts on the particle at time \(t\) seconds. The unit vectors \(\mathbf { i }\) and \(\mathbf { j }\) are directed east and north respectively.

1. Find the acceleration of the particle at time \(t\).
2. When \(t = 4\), the velocity of the particle is \(( - 3 \mathbf { i } + 56 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\). Find the velocity of the particle at time \(t\).
3. Find \(t\) when the particle is moving due west.
4. Find the speed of the particle when it is moving due west.
   
   \includegraphics[max width=\textwidth, alt={}]{3ffa0a2b-aa7d-46eb-b92b-3e3ee59f235c-09_2484_1709_223_153}

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

1. Show that at time \(t\) seconds the velocity of \(P\) is \(\left(\frac{3}{t+1} - 1\right)\) m s\(^{-1}\). [5]
2. Find, to 3 significant figures, the distance of \(P\) from \(O\) when \(P\) is instantaneously at rest. [7]

The acceleration \(a\) ms\(^{-2}\) of a particle \(P\) moving in a straight line away from a fixed point \(O\) is given by \(a = \frac{k}{1+t}\), where \(t\) is the time that has elapsed since \(P\) left \(O\), and \(k\) is a constant.

1. By solving a suitable differential equation, find an expression for the velocity \(v\) ms\(^{-1}\) of \(P\) in terms of \(t\), \(k\) and another constant \(c\). [3 marks]

Given that \(v = 0\) when \(t = 0\) and that \(v = 4\) when \(t = 2\),

1. show that \(v \ln 3 = 4 \ln (1 + t)\). [3 marks]
2. Calculate the time when \(P\) has a speed of 8 ms\(^{-1}\). [3 marks]

\(O\) is a fixed point on a horizontal plane. A particle \(P\) of mass \(0.25\) kg is released from rest at \(O\) and moves in a straight line on the plane. At time \(t\) s after release the only horizontal force acting on \(P\) has magnitude $$\frac{1}{2400}(144 - t^2) \text{ N} \quad \text{for } 0 \leqslant t \leqslant 12$$ and $$\frac{1}{2400}(t^2 - 144) \text{ N} \quad \text{for } t \geqslant 12.$$ The force acts in the direction of \(P\)'s motion. \(P\)'s velocity at time \(t\) s is \(v\) m s\(^{-1}\).

1. Find an expression for \(v\) in terms of \(t\), valid for \(t \geqslant 12\), and hence show that \(v\) is three times greater when \(t = 24\) than it is when \(t = 12\). [8]
2. Sketch the \((t, v)\) graph for \(0 \leqslant t \leqslant 24\). [3]