3.02f Non-uniform acceleration: using differentiation and integration

2. At time \(t = 0\) a particle \(P\) leaves the origin \(O\) and moves along the \(x\)-axis. At time \(t\) seconds the velocity of \(P\) is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), where $$v = 8 t - t ^ { 2 }$$

1. Find the maximum value of \(v\).
2. Find the time taken for \(P\) to return to \(O\).

1. \hspace{0pt} [In this question \(\mathbf { i }\) and \(\mathbf { j }\) are perpendicular unit vectors in a horizontal plane.]

A particle \(P\) moves in such a way that its velocity \(\mathbf { v } \mathrm { m } \mathrm { s } ^ { - 1 }\) at time \(t\) seconds is given by $$\mathbf { v } = \left( 3 t ^ { 2 } - 1 \right) \mathbf { i } + \left( 4 t - t ^ { 2 } \right) \mathbf { j }$$

1. Find the magnitude of the acceleration of \(P\) when \(t = 1\) Given that, when \(t = 0\), the position vector of \(P\) is i metres,
2. find the position vector of \(P\) when \(t = 3\)

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 \(\mathbf { i }\).

1. A particle \(P\) moves on the positive \(x\)-axis. The velocity of \(P\) at time \(t\) seconds is \(\left( 2 t ^ { 2 } - 9 t + 4 \right) \mathrm { m } \mathrm { s } ^ { - 1 }\). When \(t = 0 , P\) is 15 m from the origin \(O\).

Find

1. the values of \(t\) when \(P\) is instantaneously at rest,
2. the acceleration of \(P\) when \(t = 5\)
3. the total distance travelled by \(P\) in the interval \(0 \leqslant t \leqslant 5\)

1. At time \(t = 0\) a particle \(P\) leaves the origin \(O\) and moves along the \(x\)-axis. At time \(t\) seconds, the velocity of \(P\) is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) in the positive \(x\) direction, where

$$v = 3 t ^ { 2 } - 16 t + 21$$ The particle is instantaneously at rest when \(t = t _ { 1 }\) and when \(t = t _ { 2 } \left( t _ { 1 } < t _ { 2 } \right)\).

1. Find the value of \(t _ { 1 }\) and the value of \(t _ { 2 }\).
2. Find the magnitude of the acceleration of \(P\) at the instant when \(t = t _ { 1 }\).
3. Find the distance travelled by \(P\) in the interval \(t _ { 1 } \leqslant t \leqslant t _ { 2 }\).
4. Show that \(P\) does not return to \(O\).

1. A particle \(P\) of mass 0.5 kg moves under the action of a single force \(\mathbf { F }\) newtons. At time \(t\) seconds, \(t \geqslant 0 , P\) has velocity \(\mathbf { v } \mathrm { m } \mathrm { s } ^ { - 1 }\), where

$$\mathbf { v } = \left( 4 t - 3 t ^ { 2 } \right) \mathbf { i } + \left( t ^ { 2 } - 8 t - 40 \right) \mathbf { j }$$

1. Find
   1. the magnitude of \(\mathbf { F }\) when \(t = 3\)
   2. the acceleration of \(P\) at the instant when it is moving in the direction of the vector \(- \mathbf { i } - \mathbf { j }\). When \(t = 1 , P\) is at the point \(A\). When \(t = 2 , P\) is at the point \(B\).
2. Find, in terms of \(\mathbf { i }\) and \(\mathbf { j }\), the vector \(\overrightarrow { A B }\).

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.

3. At time \(t\) seconds, a particle \(P\) has position vector \(\mathbf { r }\) metres relative to a fixed origin \(O\), where $$\mathbf { r } = \left( t ^ { 3 } - 3 t \right) \mathbf { i } + 4 t ^ { 2 } \mathbf { j } , t \geq 0$$ Find

1. the velocity of \(P\) at time \(t\) seconds,
2. the time when \(P\) is moving parallel to the vector \(\mathbf { i } + \mathbf { j }\).
   (5)

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.

1. A spacecraft \(S\) of mass \(m\) moves in a straight line towards the centre, \(O\), of a planet.

The planet is modelled as a fixed sphere of radius \(R\).
The spacecraft \(S\) is modelled as a particle.
The gravitational force of the planet is the only force acting on \(S\).
When \(S\) is a distance \(x ( x \geqslant R )\) from \(O\)

• the gravitational force is directed towards \(O\) and has magnitude \(\frac { m g R ^ { 2 } } { 2 x ^ { 2 } }\)
• the speed of \(S\) is \(v\)
  1. Show that

$$v ^ { 2 } = \frac { g R ^ { 2 } } { x } + C$$ where \(C\) is a constant. When \(x = 3 R , v = \sqrt { 3 g R }\)

Find, in terms of \(g\) and \(R\), the speed of \(S\) as it hits the surface of the planet.

2. In this question solutions relying on calculator technology are not acceptable. A particle \(P\) of mass 2 kg is moving along the positive \(x\)-axis.
At time \(t\) seconds, where \(t \geqslant 0 , P\) is \(x\) metres from the origin \(O\) and is moving away from \(O\) with speed \(v \mathrm {~ms} ^ { - 1 }\) where \(v = \frac { 1 } { \sqrt { ( 2 x + 1 ) } }\)

1. Find the magnitude of the resultant force acting on \(P\) when its speed is \(\frac { 1 } { 3 } \mathrm {~ms} ^ { - 1 }\) When \(t = 0 , P\) is at \(O\)
2. Find the value of \(t\) when \(P\) is 7.5 m from \(O\)

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.

   VIIIV SIHI NI JIIIM ION OC

   VIIV SIHI NI JAHM ION OO

   VI4V SIHI NI JIIIM I ON OO

4. \begin{figure}[h] \end{figure} A particle \(P\) of mass \(m\) is attached to one end of a light inextensible string of length \(a\). The other end of the string is attached to a point \(O\). The point \(A\) is vertically below \(O\), and \(O A = a\). The particle is projected horizontally from \(A\) with speed \(\sqrt { } ( 3 a g )\). When \(O P\) makes an angle \(\theta\) with the upward vertical through \(O\) and the string is still taut, the tension in the string is \(T\) and the speed of \(P\) is \(v\), as shown in Figure 2.

1. Find, in terms of \(a , g\) and \(\theta\), an expression for \(v ^ { 2 }\).
2. Show that \(T = ( 1 - 3 \cos \theta ) m g\). The string becomes slack when \(P\) is at the point \(B\).
3. Find, in terms of \(a\), the vertical height of \(B\) above \(A\). After the string becomes slack, the highest point reached by \(P\) is \(C\).
4. Find, in terms of \(a\), the vertical height of \(C\) above \(B\).

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 particle \(P\) of mass \(m \mathrm {~kg}\) slides from rest down a smooth plane inclined at \(30 ^ { \circ }\) to the horizontal. When \(P\) has moved a distance \(x\) metres down the plane, the resistance to the motion of \(P\) from non-gravitational forces has magnitude \(m x ^ { 2 }\) newtons. Find

1. the speed of \(P\) when \(x = 2\),
2. the distance \(P\) has moved when it comes to rest for the first time.

7. \begin{figure}[h] \end{figure} A particle \(P\) of mass \(5 m\) is attached to one end of a light inextensible string of length \(a\). The other end of the string is attached to a fixed point \(O\). The particle is held at the point \(A\), where \(O A = a\) and \(O A\) is horizontal, as shown in Figure 6. The particle is projected vertically downwards with speed \(\sqrt { } \left( \frac { 9 a g } { 5 } \right)\). When the string makes an angle \(\theta\) with the downward vertical through \(O\) and the string is still taut, the tension in the string is \(T\).

1. Show that \(T = 3 m g ( 5 \cos \theta + 3 )\). At the instant when the particle reaches the point \(B\) the string becomes slack.
2. Find the speed of \(P\) at \(B\). At time \(t = 0 , P\) is at \(B\). At time \(t\), before the string becomes taut once more, the coordinates of \(P\) are \(( x , y )\) referred to horizontal and vertical axes with origin \(O\). The \(x\)-axis is directed along \(O A\) produced and the \(y\)-axis is vertically upward.
3. Find
   1. \(x\) in terms of \(t , a\) and \(g\),
   2. \(y\) in terms of \(t , a\) and \(g\).

3.At time \(t\) seconds( \(t \geqslant 0\) )a particle \(P\) has velocity \(\mathbf { v } \mathrm { ms } ^ { - 1 }\) ,where When \(t = 0\) the particle \(P\) is at the origin \(O\) .At time \(T\) seconds,\(P\) is at the point \(A\) and \(\mathbf { v } = \lambda ( \mathbf { i } + \mathbf { j } )\) ,where \(\lambda\) is a constant. Find

1. the value of \(T\) ,
2. the acceleration of \(P\) as it passes through the point \(A\) ,
3. the distance \(O A\) . $$\mathbf { v } = \left( 6 t ^ { 2 } + 6 t \right) \mathbf { i } + \left( 3 t ^ { 2 } + 24 \right) \mathbf { j }$$ 的 When \(t = 0\) the particle \(P\) is at the origin \(O\) .At time \(T\) seconds,\(P\) is at the point \(A\) and \(\mathbf { v } = \lambda ( \mathbf { i } + \mathbf { j } )\) ,where \(\lambda\) is a constant. Find
   1. the value of \(T\) , \(\_\_\_\_\) "

6 The velocity \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) of a particle at time \(t \mathrm {~s}\) is given by \(v = t ^ { 2 } - 9\). The particle travels in a straight line and passes through a fixed point \(O\) when \(t = 2\).

1. Find the displacement of the particle from \(O\) when \(t = 0\).
2. Calculate the distance the particle travels from its position at \(t = 0\) until it changes its direction of motion.
3. Calculate the distance of the particle from \(O\) when the acceleration of the particle is \(10 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).

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 .

5 \includegraphics[max width=\textwidth, alt={}, center]{2b3457b6-1fe9-4e67-91d4-a8bc4a5b1709-3_394_789_251_639} The diagram shows the ( \(t , v\) ) graph of an athlete running in a straight line on a horizontal track in a 100 m race. He starts from rest and has constant acceleration until he reaches a speed of \(15 \mathrm {~ms} ^ { - 1 }\) when \(t = T\). He maintains this constant speed until he decelerates at a constant rate of \(1.75 \mathrm {~ms} ^ { - 2 }\) for the final 4 s of the race. He completes the race in 10 s .

1. Calculate \(T\). The athlete races against a robot which has a displacement from the starting line of \(\left( 3 t ^ { 2 } - 0.2 t ^ { 3 } \right) \mathrm { m }\), at time \(t \mathrm {~s}\) after the start of the race.
2. Show that the speed of the robot is \(15 \mathrm {~ms} ^ { - 1 }\) when \(t = 5\).
3. Find the value of \(t\) for which the decelerations of the robot and the athlete are equal.
4. Verify that the athlete and the robot reach the finish line simultaneously.

2 A particle \(P\) moves in a straight line. The displacement of \(P\) from a fixed point on the line is \(\left( t ^ { 4 } - 2 t ^ { 3 } + 5 \right) \mathrm { m }\), where \(t\) is the time in seconds. Show that, when \(t = 1.5\),

1. \(P\) is at instantaneous rest,
2. the acceleration of \(P\) is \(9 \mathrm {~ms} ^ { - 2 }\).