Collision or meeting problems

5 Particles \(X\) and \(Y\) move in a straight line through points \(A\) and \(B\). Particle \(X\) starts from rest at \(A\) and moves towards \(B\). At the same instant, \(Y\) starts from rest at \(B\). At time \(t\) seconds after the particles start moving

• the acceleration of \(X\) in the direction \(A B\) is given by \(( 12 t + 12 ) \mathrm { m } \mathrm { s } ^ { - 2 }\),
• the acceleration of \(Y\) in the direction \(A B\) is given by \(( 24 t - 8 ) \mathrm { m } \mathrm { s } ^ { - 2 }\).
  1. It is given that the velocities of \(X\) and \(Y\) are equal when they collide.

Calculate the distance \(A B\).

It is given instead that \(A B = 36 \mathrm {~m}\). Verify that \(X\) and \(Y\) collide after 3 s.

6 Two particles \(A\) and \(B\) start to move at the same instant from a point \(O\). The particles move in the same direction along the same straight line. The acceleration of \(A\) at time \(t \mathrm {~s}\) after starting to move is \(a \mathrm {~m} \mathrm {~s} ^ { - 2 }\), where \(a = 0.05 - 0.0002 t\).

1. Find A's velocity when \(t = 200\) and when \(t = 500\).
   \(B\) moves with constant acceleration for the first 200 s and has the same velocity as \(A\) when \(t = 200 . B\) moves with constant retardation from \(t = 200\) to \(t = 500\) and has the same velocity as \(A\) when \(t = 500\).
2. Find the distance between \(A\) and \(B\) when \(t = 500\).

7 Particles \(P\) and \(Q\) leave a fixed point \(A\) at the same time and travel in the same straight line. The velocity of \(P\) after \(t\) seconds is \(6 t ( t - 3 ) \mathrm { m } \mathrm { s } ^ { - 1 }\) and the velocity of \(Q\) after \(t\) seconds is \(( 10 - 2 t ) \mathrm { m } \mathrm { s } ^ { - 1 }\).

1. Sketch, on the same axes, velocity-time graphs for \(P\) and \(Q\) for \(0 \leqslant t \leqslant 5\).
2. Verify that \(P\) and \(Q\) meet after 5 seconds.
3. Find the greatest distance between \(P\) and \(Q\) for \(0 \leqslant t \leqslant 5\).
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

4 Particles \(P\) and \(Q\) move on a straight line \(A O B\). The particles leave \(O\) simultaneously, with \(P\) moving towards \(A\) and with \(Q\) moving towards \(B\). The initial speed of \(P\) is \(1.3 \mathrm {~ms} ^ { - 1 }\) and its acceleration in the direction \(O A\) is \(0.1 \mathrm {~m} \mathrm {~s} ^ { - 2 } . Q\) moves with acceleration in the direction \(O B\) of \(0.016 t \mathrm {~m} \mathrm {~s} ^ { - 2 }\), where \(t\) seconds is the time elapsed since the instant that \(P\) and \(Q\) started to move from \(O\). When \(t = 20\), particle \(P\) passes through \(A\) and particle \(Q\) passes through \(B\).

1. Given that the speed of \(Q\) at \(B\) is the same as the speed of \(P\) at \(A\), find the speed of \(Q\) at time \(t = 0\).
2. Find the distance \(A B\).

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.

7 A particle \(P\) is projected from a fixed point \(O\) on a straight line. The displacement \(x\) m of \(P\) from \(O\) at time \(t \mathrm {~s}\) after projection is given by \(x = 0.1 t ^ { 3 } - 0.3 t ^ { 2 } + 0.2 t\).

1. Express the velocity and acceleration of \(P\) in terms of \(t\).
2. Show that when the acceleration of \(P\) is zero, \(P\) is at \(O\).
3. Find the values of \(t\) when \(P\) is stationary. At the instant when \(P\) first leaves \(O\), a particle \(Q\) is projected from \(O\). \(Q\) moves on the same straight line as \(P\) and at time \(t \mathrm {~s}\) after projection the velocity of \(Q\) is given by \(\left( 0.2 t ^ { 2 } - 0.4 \right) \mathrm { ms } ^ { - 1 } . P\) and \(Q\) collide first when \(t = T\).
4. Show that \(T\) satisfies the equation \(t ^ { 2 } - 9 t + 18 = 0\), and hence find \(T\).

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.

7
\includegraphics[max width=\textwidth, alt={}, center]{c6bac5bf-960e-4c3d-b9fa-c52de66ba719-4_652_1068_255_500} The diagram shows the ( \(t , v\) ) graphs for two particles \(A\) and \(B\) which move on the same straight line. The units of \(v\) and \(t\) are \(\mathrm { ms } ^ { - 1 }\) and s respectively. Both particles are at the point \(S\) on the line when \(t = 0\). The particle \(A\) is initially at rest, and moves with acceleration \(0.18 t \mathrm {~ms} ^ { - 2 }\) until the two particles collide when \(t = 16\). The initial velocity of \(B\) is \(U \mathrm {~ms} ^ { - 1 }\) and \(B\) has variable acceleration for the first five seconds of its motion. For the next ten seconds of its motion \(B\) has a constant velocity of \(9 \mathrm {~ms} ^ { - 1 }\); finally \(B\) moves with constant deceleration for one second before it collides with \(A\).

1. Calculate the value of \(t\) at which the two particles have the same velocity. For \(0 \leqslant t \leqslant 5\) the distance of \(B\) from \(S\) is \(\left( U t + 0.08 t ^ { 3 } \right) \mathrm { m }\).
2. Calculate \(U\) and verify that when \(t = 5 , B\) is 25 m from \(S\).
3. Calculate the velocity of \(B\) when \(t = 16\). \section*{END OF QUESTION PAPER}

7 Two particles, \(A\) and \(B\), move on a horizontal surface with constant accelerations of \(- 0.4 \mathbf { i } \mathrm {~m} \mathrm {~s} ^ { - 2 }\) and \(0.2 \mathbf { j } \mathrm {~m} \mathrm {~s} ^ { - 2 }\) respectively. At time \(t = 0\), particle \(A\) starts at the origin with velocity \(( 4 \mathbf { i } + 2 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\). At time \(t = 0\), particle \(B\) starts at the point with position vector \(11.2 \mathbf { i }\) metres, with velocity \(( 0.4 \mathbf { i } + 0.6 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\).

1. Find the position vector of \(A , 10\) seconds after it leaves the origin.
   [0pt] [2 marks]
2. Show that the two particles collide, and find the position vector of the point where they collide.
   [0pt] [9 marks]
   \includegraphics[max width=\textwidth, alt={}]{788534a5-abbb-4d6a-87b2-c54e859a128a-16_1881_1707_822_153}
   \includegraphics[max width=\textwidth, alt={}]{788534a5-abbb-4d6a-87b2-c54e859a128a-17_2484_1707_221_153}

5 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 \(+ 9 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) 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 \(t = 2\). \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{b9e41fac-9f4b-4165-af03-67ebdcb326de-3_349_987_375_623} \captionsetup{labelformat=empty} \caption{Fig. 4}
   \end{figure} Particles P and Q move in the same straight line. Particle P starts from rest and has a constant acceleration towards \(Q\) of \(0.5 \mathrm {~m} \mathrm {~s} ^ { - 2 }\). Particle \(Q\) starts 125 m from \(P\) at the same time and has a constant speed of \(10 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) away from \(P\). The initial values are shown in Fig. 4.
3. Write down expressions for the distances travelled by P and by Q at time \(t\) seconds after the start of the motion.
4. How much time does it take for P to catch up with Q and how far does P travel in this time?

9 Two particles \(A\) and \(B\) have position vectors \(\mathbf { r } _ { A }\) metres and \(\mathbf { r } _ { B }\) metres at time \(t\) seconds, where $$\mathbf { r } _ { A } = t ^ { 2 } \mathbf { i } + ( 3 t - 1 ) \mathbf { j } \quad \text { and } \quad \mathbf { r } _ { B } = \left( 1 - 2 t ^ { 2 } \right) \mathbf { i } + \left( 3 t - 2 t ^ { 2 } \right) \mathbf { j } , \quad \text { for } t \geqslant 0$$

1. Find the values of \(t\) when \(A\) and \(B\) are moving with the same speed.
2. Show that the distance, \(d\) metres, between \(A\) and \(B\) at time \(t\) satisfies $$d ^ { 2 } = 13 t ^ { 4 } - 10 t ^ { 2 } + 2$$
3. Hence find the shortest distance between \(A\) and \(B\) in the subsequent motion.