3.02g Two-dimensional variable acceleration

1. Particle \(P\) has mass \(m\) and particle \(Q\) has mass \(5 m\).

The particles are moving in the same direction along the same straight line on a smooth horizontal surface. Particle \(P\) collides directly with particle \(Q\).
Immediately before the collision, the speed of \(P\) is \(6 u\) and the speed of \(Q\) is \(u\).
Immediately after the collision, the speed of \(P\) is \(x\) and the speed of \(Q\) is \(y\).
The direction of motion of \(P\) is reversed as a result of the collision.
The coefficient of restitution between \(P\) and \(Q\) is \(e\).

1. Find the complete range of possible values of \(e\). Given that \(e = \frac { 3 } { 5 }\)
2. find the total kinetic energy lost in the collision between \(P\) and \(Q\). After the collision, \(Q\) hits a smooth fixed vertical wall that is perpendicular to the direction of motion of \(Q\). Particle \(Q\) rebounds.
   The coefficient of restitution between \(Q\) and the wall is \(f\).
   Given that there is a second collision between \(P\) and \(Q\),
3. find the complete range of possible values of \(f\).

1. \hspace{0pt} [In this question \(\mathbf { i }\) and \(\mathbf { j }\) are unit vectors, with \(\mathbf { i }\) horizontal and \(\mathbf { j }\) vertical.]

\begin{figure}[h] \end{figure} The fixed points \(A\) and \(B\) lie on horizontal ground.
At time \(t = 0\), a particle \(P\) is projected from \(A\) with velocity \(( 4 \mathbf { i } + 7 \mathbf { j } ) \mathrm { ms } ^ { - 1 }\) Particle \(P\) moves freely under gravity and hits the ground at \(B\), as shown in Figure 6 .

1. Find the distance \(A B\). The speed of \(P\) is less than \(5 \mathrm {~ms} ^ { - 1 }\) for an interval of length \(T\) seconds.
2. Find the value of \(T\) At the instant when the direction of motion of \(P\) is perpendicular to the initial direction of motion of \(P\), the particle is \(h\) metres above the ground.
3. Find the value of \(h\).

6. [In this question the unit vectors \(\mathbf { i }\) and \(\mathbf { j }\) are in a vertical plane, \(\mathbf { i }\) being horizontal and \(\mathbf { j }\) being vertically upwards.] At \(t = 0\) a particle \(P\) is projected from a fixed point \(O\) with velocity ( \(7 \mathbf { i } + 7 \sqrt { 3 } \mathbf { j }\) ) \(\mathrm { m } \mathrm { s } ^ { - 1 }\). The particle moves freely under gravity. The position vector of a point on the path of \(P\) is \(( x \mathbf { i } + y \mathbf { j } ) \mathrm { m }\) relative to \(O\).

1. Show that $$y = \sqrt { 3 } x - \frac { g } { 98 } x ^ { 2 }$$
2. Find the direction of motion of \(P\) when it passes through the point on the path where \(x = 20\) At time \(T\) seconds \(P\) passes through the point with position vector \(( 2 \lambda \mathbf { i } + \lambda \mathbf { j } ) \mathrm { m }\) where \(\lambda\) is a positive constant.
3. Find the value of \(T\).
   DO NOT WIRITE IN THIS AREA

3. A particle \(P\) moves along the \(x\)-axis. At time \(t = 0 , P\) passes through the origin with speed \(6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) in the positive \(x\) direction. The acceleration of \(P\) at time \(t\) seconds, where \(t \geqslant 0\), is \(( 4 t - 8 ) \mathrm { m } \mathrm { s } ^ { - 2 }\) in the positive \(x\) direction.

1. 1. Show that \(P\) is instantaneously at rest when \(t = 1\)
   2. Find the other value of \(t\) for which \(P\) is instantaneously at rest.
2. Find the total distance travelled by \(P\) in the interval \(1 \leqslant t \leqslant 4\)

5. At time \(t\) seconds ( \(t \geqslant 0\) ), a particle \(P\) has velocity \(\mathbf { v m ~ s } ^ { - 1 }\), where $$\mathbf { v } = \left( 3 t ^ { 2 } - 9 t + 6 \right) \mathbf { i } + \left( t ^ { 2 } + t - 6 \right) \mathbf { j }$$

1. Find the acceleration of \(P\) when \(t = 3\) When \(t = 0 , P\) is at the fixed point \(O\).
   The particle comes to instantaneous rest at the point \(A\).
2. Find the distance \(O A\).

1. At time \(t\) seconds, \(t \geqslant 0\), a particle \(P\) has position vector \(\mathbf { r }\) metres with respect to a fixed origin \(O\), where

$$\mathbf { r } = \left( t ^ { 3 } - 8 t \right) \mathbf { i } + \left( \frac { 1 } { 3 } t ^ { 3 } - t ^ { 2 } + 2 t \right) \mathbf { j }$$

1. Find the acceleration of \(P\) when \(t = 4\) At time \(T\) seconds, \(T \geqslant 0 , P\) is moving in the direction of ( \(2 \mathbf { i } + \mathbf { j }\) )
2. Find the value of \(T\)

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

In this question you must show all stages of your working. Solutions relying on calculator technology are not acceptable. A particle \(P\) is moving on a smooth horizontal plane.
At time \(t\) seconds \(( t \geqslant 0 )\), the position vector of \(P\), relative to a fixed point \(O\), is \(\mathbf { r }\) metres and the velocity of \(P\) is \(\mathbf { v } \mathrm { ms } ^ { - 1 }\) where $$\mathbf { v } = \left( 4 t ^ { 2 } - 5 t \right) \mathbf { i } + ( - 10 t - 12 ) \mathbf { j }$$ When \(t = 0 , \mathbf { r } = 2 \mathbf { i } + 6 \mathbf { j }\)

1. Find \(\mathbf { r }\) when \(t = 2\) When \(t = T\) particle \(P\) is moving in the direction of the vector \(\mathbf { i } - 2 \mathbf { j }\)
2. Find the value of \(T\)
3. Find the exact magnitude of the acceleration of \(P\) when \(t = 2.5\)

1. At time \(t\) seconds \(( t \geqslant 0 )\), a particle \(P\) has position vector \(\mathbf { r }\) metres with respect to a fixed origin \(O\), where

$$\mathbf { r } = \left( t ^ { 3 } - \frac { 9 } { 2 } t ^ { 2 } - 24 t \right) \mathbf { i } + \left( - t ^ { 3 } + 3 t ^ { 2 } + 12 t \right) \mathbf { j }$$ At time \(T\) seconds, \(P\) is moving in a direction parallel to the vector \(\mathbf { - i } - \mathbf { j }\) Find

1. the value of \(T\),
2. the magnitude of the acceleration of \(P\) at the instant when \(t = T\).

3. At time \(t\) seconds \(( t \geqslant 0 )\) a particle \(P\) has position vector \(\mathbf { r }\) metres, with respect to a fixed origin \(O\), where
(b) the magnitude of the acceleration of \(P\) when \(t = 4\) $$\begin{aligned} & \qquad \mathbf { r } = \left( 16 t - 3 t ^ { 3 } \right) \mathbf { i } + \left( t ^ { 3 } - t ^ { 2 } + 2 \right) \mathbf { j } \\ & \text { Find } \\ & \text { (a) the velocity of } P \text { at the instant when it is moving parallel to the vector } \mathbf { j } \text {, } \end{aligned}$$ VILIV SIHI NI IIIIIM ION OC
VILV SIHI NI JAHAM ION OC
VJ4V SIHI NI JIIYM ION OC

3. A particle \(P\) moves on the \(x\)-axis. At time \(t = 0 , P\) is instantaneously at rest at \(O\).
At time \(t\) seconds, \(t > 0\), the \(x\) coordinate of \(P\) is given by $$x = 2 t ^ { \frac { 7 } { 2 } } - 14 t ^ { \frac { 5 } { 2 } } + \frac { 56 } { 3 } t ^ { \frac { 3 } { 2 } }$$ Find

1. the non-zero values of \(t\) for which \(P\) is at instantaneous rest
2. the total distance travelled by \(P\) in the interval \(0 \leqslant t \leqslant 4\)
3. the acceleration of \(P\) when \(t = 4\) \(\_\_\_\_\)

4. At time \(t\) seconds \(( 0 \leqslant t < 5 )\), a particle \(P\) has velocity \(\mathbf { v m s } ^ { - 1 }\), where $$\mathbf { v } = ( \sqrt { 5 - t } ) \mathbf { i } + \left( t ^ { 2 } + 2 t - 3 \right) \mathbf { j }$$ When \(t = \lambda\), particle \(P\) is moving in a direction parallel to the vector \(\mathbf { i }\).

1. Find the acceleration of \(P\) when \(t = \lambda\) The position vector of \(P\) is measured relative to the fixed point \(O\) When \(t = 1\), the position vector of \(P\) is \(( - 2 \mathbf { i } + \mathbf { j } ) \mathrm { m }\). Given that \(1 \leqslant T < 5\)
2. find, in terms of \(T\), the position vector of \(P\) when \(t = T\)

1. At time \(t\) seconds, \(t > 0\), a particle \(P\) is at the point with position vector \(\mathbf { r } \mathrm { m }\), where

$$\mathbf { r } = \left( t ^ { 4 } - 8 t ^ { 2 } \right) \mathbf { i } + \left( 6 t ^ { 2 } - 2 t ^ { \frac { 3 } { 2 } } \right) \mathbf { j }$$

1. Find the velocity of \(P\) when \(P\) is moving in a direction parallel to the vector \(\mathbf { j }\)
2. Find the acceleration of \(P\) when \(t = 4\)

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.

4. At time \(t\) seconds the velocity of a particle \(P\) is \([ ( 4 t - 5 ) \mathbf { i } + 3 \mathbf { j } ] \mathrm { m } \mathrm { s } ^ { - 1 }\). When \(t = 0\), the position vector of \(P\) is \(( 2 \mathbf { i } + 5 \mathbf { j } ) \mathrm { m }\), relative to a fixed origin \(O\).

1. Find the value of \(t\) when the velocity of \(P\) is parallel to the vector \(\mathbf { j }\).
2. Find an expression for the position vector of \(P\) at time \(t\) seconds. A second particle \(Q\) moves with constant velocity \(( - 2 \mathbf { i } + c \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\). When \(t = 0\), the position vector of \(Q\) is \(( 11 \mathbf { i } + 2 \mathbf { j } ) \mathrm { m }\). The particles \(P\) and \(Q\) collide at the point with position vector ( \(d \mathbf { i } + 14 \mathbf { j }\) ) m.
3. Find
   1. the value of \(c\),
   2. the value of \(d\).

4. At time \(t\) seconds, the velocity of a particle \(P\) is \([ ( 4 t - 7 ) \mathbf { i } - 5 \mathbf { j } ] \mathrm { m } \mathrm { s } ^ { - 1 }\). When \(t = 0 , P\) is at the point with position vector \(( 3 \mathbf { i } + 5 \mathbf { j } ) \mathrm { m }\) relative to a fixed origin \(O\).

1. Find an expression for the position vector of \(P\) after \(t\) seconds, giving your answer in the form \(( a \mathbf { i } + b \mathbf { j } ) \mathrm { m }\). A second particle \(Q\) moves with constant velocity \(( 2 \mathbf { i } - 3 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\). When \(t = 0\), the position vector of \(Q\) is \(( - 7 \mathrm { i } ) \mathrm { m }\).
2. Prove that \(P\) and \(Q\) collide.

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 \(P\) 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 \(\overrightarrow { A B }\).

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)

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.A particle \(P\) moves in a horizontal plane.At time \(t\) seconds,the position vector of \(P\) is \(\mathbf { r }\) metres relative to a fixed origin \(O\) ,and \(\mathbf { r }\) is given by $$\mathbf { r } = \left( 18 t - 4 t ^ { 3 } \right) \mathbf { i } + c t ^ { 2 } \mathbf { j } ,$$ where \(c\) is a positive constant.When \(t = 1.5\) ,the speed of \(P\) is \(15 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) .Find

1. the value of \(c\) ,
2. the acceleration of \(P\) when \(t = 1.5\) . \(\mathbf { r }\) metres relative to a fixed origin \(O\) ,and \(\mathbf { r }\) is given by $$\begin{aligned} \mathbf { r } = \left( 18 t - 4 t ^ { 3 } \right) \mathbf { i } + c t ^ { 2 } \mathbf { j } , \\ \text { where } c \text { is a positive constant.When } t = 1.5 \text { ,the speed of } P \text { is } 15 \mathrm {~m} \mathrm {~s} ^ { - 1 } \text { .Find } \end{aligned}$$ (a)the value of \(c\) , 3.A particle \(P\) moves in a horizontal plane.At time \(t\) seconds,the position vector of \(P\) is D墐
   (b)the acceleration of \(P\) when \(t = 1.5\) .

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 \(T\) , \(\_\_\_\_\) "

7 \includegraphics[max width=\textwidth, alt={}, center]{f0813713-d677-4ed7-87e1-971a64bdb6ff-4_122_255_1503_561} The diagram shows two particles \(P\) and \(Q\), of masses 0.2 kg and 0.3 kg respectively, which move on a horizontal surface in the same direction along a straight line. A stationary particle \(R\) of mass 1.5 kg also lies on this line. \(P\) and \(Q\) collide and coalesce to form a combined particle \(C\). Immediately before this collision \(P\) has velocity \(4 \mathrm {~ms} ^ { - 1 }\) and \(Q\) has velocity \(2.5 \mathrm {~ms} ^ { - 1 }\).

1. Calculate the velocity of \(C\) immediately after this collision. At time \(t \mathrm {~s}\) after this collision the velocity \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) of \(C\) is given by \(v = V _ { 0 } - 3 t ^ { 2 }\) for \(0 < t \leqslant 0.3\). \(C\) strikes \(R\) when \(t = 0.3\).
2. (a) State the value of \(V _ { 0 }\).
   (b) Calculate the distance \(C\) moves before it strikes \(R\).
   (c) Find the acceleration of \(C\) immediately before it strikes \(R\). Immediately after \(C\) strikes \(R\), the particles have equal speeds but move in opposite directions.
3. Find the speed of \(C\) immediately after it strikes \(R\).

6 A particle \(P\) moves in a straight line. At time \(t\) s after passing through a point \(O\) of the line, the displacement of \(P\) from \(O\) is \(x \mathrm {~m}\). Given that \(x = 0.06 t ^ { 3 } - 0.45 t ^ { 2 } - 0.24 t\), find

1. the velocity and the acceleration of \(P\) when \(t = 0\),
2. the value of \(x\) when \(P\) has its minimum velocity, and the speed of \(P\) at this instant,
3. the positive value of \(t\) when the direction of motion of \(P\) changes.

6 A particle \(P\) moves in a straight line on a horizontal surface. \(P\) passes through a fixed point \(O\) on the line with velocity \(2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). At time \(t \mathrm {~s}\) after passing through \(O\), the acceleration of \(P\) is \(( 4 + 12 t ) \mathrm { m } \mathrm { s } ^ { - 2 }\).

1. Calculate the velocity of \(P\) when \(t = 3\).
2. Find the distance \(O P\) when \(t = 3\). A second particle \(Q\), having the same mass as \(P\), moves along the same straight line. The displacement of \(Q\) from \(O\) is \(\left( k - 2 t ^ { 3 } \right) \mathrm { m }\), where \(k\) is a constant. When \(t = 3\) the particles collide and coalesce.
3. Find the value of \(k\).
4. Find the common velocity of the particles immediately after their collision.