Displacement from velocity by integration

5 A particle \(P\) moves in a straight line that passes through the origin \(O\). The velocity of \(P\) at time \(t\) seconds is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), where \(v = 20 t - t ^ { 3 }\). At time \(t = 0\) the particle is at rest at a point whose displacement from \(O\) is - 36 m .

1. Find an expression for the displacement of \(P\) from \(O\) in terms of \(t\).
2. Find the displacement of \(P\) from \(O\) when \(t = 4\).
3. Find the values of \(t\) for which the particle is at \(O\).

2 A particle starts at a point \(O\) and moves along a straight line. Its velocity \(t\) s after leaving \(O\) is \(\left( 1.2 t - 0.12 t ^ { 2 } \right) \mathrm { m } \mathrm { s } ^ { - 1 }\). Find the displacement of the particle from \(O\) when its acceleration is \(0.6 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).

2 \includegraphics[max width=\textwidth, alt={}, center]{fcd2b219-d9b4-4972-b8fe-25cf543b9054-2_649_1244_482_452} A man runs in a straight line. He passes through a fixed point \(A\) with constant velocity \(7 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at time \(t = 0\). At time \(t \mathrm {~s}\) his velocity is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The diagram shows the graph of \(v\) against \(t\) for the period \(0 \leqslant t \leqslant 40\).

1. Show that the man runs more than 154 m in the first 24 s .
2. Given that the man runs 20 m in the interval \(20 \leqslant t \leqslant 24\), find how far he is from \(A\) when \(t = 40\).

2 A particle moves in a straight line. Its velocity \(t\) seconds after leaving a fixed point \(O\) on the line is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), where \(v = 0.2 t + 0.006 t ^ { 2 }\). For the instant when the acceleration of the particle is 2.5 times its initial acceleration,

1. show that \(t = 25\),
2. find the displacement of the particle from \(O\).

5 A particle starts from a fixed origin with velocity \(0.4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and moves in a straight line. The acceleration \(a \mathrm {~m} \mathrm {~s} ^ { - 2 }\) of the particle \(t \mathrm {~s}\) after it leaves the origin is given by \(a = k \left( 3 t ^ { 2 } - 12 t + 2 \right)\), where \(k\) is a constant. When \(t = 1\), the velocity of \(P\) is \(0.1 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

1. Show that the value of \(k\) is 0.1 .
2. Find an expression for the displacement of the particle from the origin in terms of \(t\).
3. Hence verify that the particle is again at the origin at \(t = 2\).

3 A particle \(P\) is moving in a horizontal straight line. Initially \(P\) is at the point \(O\) on the line and is moving with velocity \(25 \mathrm {~ms} ^ { - 1 }\). At time \(t \mathrm {~s}\) after passing through \(O\), the acceleration of \(P\) is \(\frac { 4000 } { ( 5 t + 4 ) ^ { 3 } } \mathrm {~ms} ^ { - 2 }\) in the direction \(P O\). The displacement of \(P\) from \(O\) at time \(t\) is \(x \mathrm {~m}\). Find an expression for \(x\) in terms of \(t\). \includegraphics[max width=\textwidth, alt={}, center]{c486c59a-2493-4dd3-bf1e-dde57fe744d9-06_894_809_260_628} An object is composed of a hemispherical shell of radius \(2 a\) attached to a closed hollow circular cylinder of height \(h\) and base radius \(a\). The hemispherical shell and the hollow cylinder are made of the same uniform material. The axes of symmetry of the shell and the cylinder coincide. \(A B\) is a diameter of the lower end of the cylinder (see diagram).

1. Find, in terms of \(a\) and \(h\), an expression for the distance of the centre of mass of the object from \(A B\). [4]
   The object is placed on a rough plane which is inclined to the horizontal at an angle \(\theta\), where \(\tan \theta = \frac { 2 } { 3 }\). The object is in equilibrium with \(A B\) in contact with the plane and lying along a line of greatest slope of the plane.
2. Find the set of possible values of \(h\), in terms of \(a\). \includegraphics[max width=\textwidth, alt={}, center]{c486c59a-2493-4dd3-bf1e-dde57fe744d9-08_629_1358_269_367} A light inextensible string \(A B\) passes through two small holes \(C\) and \(D\) in a smooth horizontal table where \(A C = 3 a\) and \(D B = a\). A particle of mass \(m\) is attached at the end \(A\) and moves in a horizontal circle with angular velocity \(\omega\). A particle of mass \(\frac { 3 } { 4 } m\) is attached to the end \(B\) and moves in a horizontal circle with angular velocity \(k \omega\). \(A C\) makes an angle \(\theta\) with the downward vertical and \(D B\) makes an angle \(\theta\) with the horizontal (see diagram). Find the value of \(k\).

5. A particle moves along the \(x\)-axis. At time \(t\) seconds, \(t \geqslant 0\), the velocity of the particle is \(v \mathrm {~ms} ^ { - 1 }\) in the direction of \(x\) increasing, where \(v = 2 t ^ { \frac { 3 } { 2 } } - 6 t + 2\) At time \(t = 0\) the particle passes through the origin \(O\). At the instant when the acceleration of the particle is zero, the particle is at the point \(A\). Find the distance \(O A\).
(8)

3 Fig. 3 is a sketch of the velocity-time graph modelling the velocity of a sprinter at the start of a race. \begin{figure}[h] \end{figure}

1. How can you tell from the sketch that the acceleration is not modelled as being constant for \(0 \leqslant t \leqslant 4\) ? The velocity of the sprinter, \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), for the time interval \(0 \leqslant t \leqslant 4\) is modelled by the expression $$v = 3 t - \frac { 3 } { 8 } t ^ { 2 }$$
2. Find the acceleration that the model predicts for \(t = 4\) and comment on what this suggests about the running of the sprinter.
3. Calculate the distance run by the sprinter from \(t = 1\) to \(t = 4\).

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

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)

6 A swimmer \(C\) swims with velocity \(v \mathrm {~ms} ^ { - 1 }\) in a swimming pool. At time \(t \mathrm {~s}\) after starting, \(v = 0.006 t ^ { 2 } - 0.18 t + k\), where \(k\) is a constant. \(C\) swims from one end of the pool to the other in 28.4 s .

1. Find the acceleration of \(C\) in terms of \(t\).
2. Given that the minimum speed of \(C\) is \(0.65 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), show that \(k = 2\).
3. Express the distance travelled by \(C\) in terms of \(t\), and calculate the length of the pool.

3 A particle \(P\) travels in a straight line. The velocity of \(P\) at time \(t\) seconds after it passes through a fixed point \(A\) is given by \(\left( 0.6 t ^ { 2 } + 3 \right) \mathrm { ms } ^ { - 1 }\). Find

1. the velocity of \(P\) when it passes through \(A\),
2. the displacement of \(P\) from \(A\) when \(t = 1.5\),
3. the velocity of \(P\) when it has acceleration \(6 \mathrm {~ms} ^ { - 2 }\).

1. A particle \(P\) moves along a straight line such that at time \(t\) seconds, \(t \geqslant 0\), after leaving the point \(O\) on the line, the velocity, \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), of \(P\) is modelled as

$$v = ( 7 - 2 t ) ( t + 2 )$$

1. Find the value of \(t\) at the instant when \(P\) stops accelerating.
2. Find the distance of \(P\) from \(O\) at the instant when \(P\) changes its direction of motion. In this question, solutions relying on calculator technology are not acceptable.

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

A particle is moving along a straight line.
At time t seconds, \(\mathrm { t } > 0\), the velocity of the particle is \(\mathrm { Vms } ^ { - 1 }\), where $$v = 2 t - 7 \sqrt { t } + 6$$

1. Find the acceleration of the particle when \(t = 4\) When \(\mathrm { t } = 1\) the particle is at the point X .
   When \(\mathrm { t } = 2\) the particle is at the point Y . Given that the particle does not come to instantaneous rest in the interval \(1 < \mathrm { t } < 2\)
2. show that \(X Y = \frac { 1 } { 3 } ( 41 - 28 \sqrt { 2 } )\) metres.

4 A particle moves along a straight line through an origin O . Initially the particle is at O .
At time \(t \mathrm {~s}\), its displacement from O is \(x \mathrm {~m}\) and its velocity, \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), is given by $$v = 24 - 18 t + 3 t ^ { 2 }$$

1. Find the times, \(T _ { 1 } \mathrm {~s}\) and \(T _ { 2 } \mathrm {~s}\) (where \(T _ { 2 } > T _ { 1 }\) ), at which the particle is stationary.
2. Find an expression for \(x\) at time \(t\) s. Show that when \(t = T _ { 1 } , x = 20\) and find the value of \(x\) when \(t = T _ { 2 }\).

5 Fig. 3 is a sketch of the velocity-time graph modelling the velocity of a sprinter at the start of a race. \begin{figure}[h] \end{figure}

1. How can you tell from the sketch that the acceleration is not modelled as being constant for \(0 \leqslant t \leqslant 4\) ? The velocity of the sprinter, \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), for the time interval \(0 \leqslant t \leqslant 4\) is modelled by the expression $$v = 3 t - \frac { 3 } { 8 } t ^ { 2 } .$$
2. Find the acceleration that the model predicts for \(t = 4\) and comment on what this suggests about the running of the sprinter.
3. Calculate the distance run by the sprinter from \(t = 1\) to \(t = 4\).

5 Fig. 3 is a sketch of the velocity-time graph modelling the velocity of a sprinter at the start of a race. \begin{figure}[h] \end{figure}

1. How can you tell from the sketch that the acceleration is not modelled as being constant for \(0 \leqslant t \leqslant 4\) ? The velocity of the sprinter, \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), for the time interval \(0 \leqslant t \leqslant 4\) is modelled by the expression $$v = 3 t - \frac { 3 } { 8 } t ^ { 2 } .$$
2. Find the acceleration that the model predicts for \(t = 4\) and comment on what this suggests about the running of the sprinter.
3. Calculate the distance run by the sprinter from \(t = 1\) to \(t = 4\).

3 A particle moves in a straight line and at time \(t\) has velocity \(v\), where $$v = 2 t - 12 \mathrm { e } ^ { - t } , \quad t \geqslant 0$$

1. 1. Find an expression for the acceleration of the particle at time \(t\).
   2. State the range of values of the acceleration of the particle.
2. When \(t = 0\), the particle is at the origin. Find an expression for the displacement of the particle from the origin at time \(t\).

2 A particle moves in a straight line and at time \(t\) it has velocity \(v\), where $$v = 3 t ^ { 2 } - 2 \sin 3 t + 6$$

1. 1. Find an expression for the acceleration of the particle at time \(t\).
   2. When \(t = \frac { \pi } { 3 }\), show that the acceleration of the particle is \(2 \pi + 6\).
2. When \(t = 0\), the particle is at the origin. Find an expression for the displacement of the particle from the origin at time \(t\).

2 A particle moves in a straight line. At time \(t\) seconds, it has velocity \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), where $$v = 6 t ^ { 2 } - 2 \mathrm { e } ^ { - 4 t } + 8$$ and \(t \geqslant 0\).

1. 1. Find an expression for the acceleration of the particle at time \(t\).
   2. Find the acceleration of the particle when \(t = 0.5\).
2. The particle has mass 4 kg . Find the magnitude of the force acting on the particle when \(t = 0.5\).
3. When \(t = 0\), the particle is at the origin. Find an expression for the displacement of the particle from the origin at time \(t\).

1. A particle \(P\) moves on the \(x\)-axis. At time \(t\) seconds, \(t \geqslant 0 , P\) is \(x\) metres from the origin \(O\) and moving with velocity \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) in the direction of \(x\) increasing, where

$$v = 5 \sin 2 t$$ When \(t = 0 , x = 1\) and \(P\) is at rest.

1. Find the magnitude and direction of the acceleration of \(P\) at the instant when \(P\) is next at rest.
2. Show that \(1 \leqslant x \leqslant 6\)
3. Find the total time, in the first \(4 \pi\) seconds of the motion, for which \(P\) is more than 3 metres from \(O\)
   \includegraphics[max width=\textwidth, alt={}]{a7901165-1679-4d30-9444-0c27020e32ea-16_2260_52_309_1982}

1 A particle moves along a straight line. At time \(t\), it has velocity \(v\), where $$v = 4 t ^ { 3 } - 8 \sin 2 t + 5$$ When \(t = 0\), the particle is at the origin.
Find an expression for the displacement of the particle from the origin at time \(t\).

5 A golf ball, of mass \(m \mathrm {~kg}\), is moving in a straight line across smooth horizontal ground. At time \(t\) seconds, the golf ball has speed \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\). As the golf ball moves, it experiences a resistance force of magnitude \(0.2 m v ^ { \frac { 1 } { 2 } }\) newtons until it comes to rest. No other horizontal force acts on the golf ball. Model the golf ball as a particle.

1. Show that $$\frac { \mathrm { d } v } { \mathrm {~d} t } = - 0.2 v ^ { \frac { 1 } { 2 } }$$
2. When \(t = 0\), the speed of the golf ball is \(16 \mathrm {~ms} ^ { - 1 }\). Show that \(v = ( 4 - 0.1 t ) ^ { 2 }\).
3. Find the value of \(t\) when \(v = 1\).
4. Find the distance travelled by the golf ball as its speed decreases from \(16 \mathrm {~ms} ^ { - 1 }\) to \(1 \mathrm {~ms} ^ { - 1 }\).

A particle \(P\) moves in a straight line. At time \(t\) s after leaving a point \(O\) on the line, \(P\) has velocity \(v\text{ ms}^{-1}\), where \(v = 44t - 6t^2 - 36\).

1. Find the set of values of \(t\) for which the acceleration of the particle is positive. [2]
2. Find the two values of \(t\) at which \(P\) returns to \(O\). [3]