3.02a Kinematics language: position, displacement, velocity, acceleration

6 A particle travels in a straight line \(P Q\). The velocity of the particle \(t \mathrm {~s}\) after leaving \(P\) is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), where $$v = 4.5 + 4 t - 0.5 t ^ { 2 }$$

1. Find the velocity of the particle at the instant when its acceleration is zero.
   The particle comes to instantaneous rest at \(Q\).
2. Find the distance \(P Q\). \includegraphics[max width=\textwidth, alt={}, center]{55090630-1413-45cd-8201-4d58662db6bd-10_625_780_260_744} Two particles \(A\) and \(B\), of masses \(3 m \mathrm {~kg}\) and \(2 m \mathrm {~kg}\) respectively, are attached to the ends of a light inextensible string. The string passes over a fixed smooth pulley which is attached to the edge of a plane. The plane is inclined at an angle \(\theta\) to the horizontal. \(A\) lies on the plane and \(B\) hangs vertically, 0.8 m above the floor, which is horizontal. The string between \(A\) and the pulley is parallel to a line of greatest slope of the plane (see diagram). Initially \(A\) and \(B\) are at rest.
   1. Given that the plane is smooth, find the value of \(\theta\) for which \(A\) remains at rest.
      It is given instead that the plane is rough, \(\theta = 30 ^ { \circ }\) and the acceleration of \(A\) up the plane is \(0.1 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).
   2. Show that the coefficient of friction between \(A\) and the plane is \(\frac { 1 } { 10 } \sqrt { 3 }\).
   3. When \(B\) reaches the floor it comes to rest. Find the length of time after \(B\) reaches the floor for which \(A\) is moving up the plane. [You may assume that \(A\) does not reach the pulley.]
      If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

6 A particle moves in a straight line and passes through the point \(A\) at time \(t = 0\). The velocity of the particle at time \(t \mathrm {~s}\) after leaving \(A\) is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), where $$v = 2 t ^ { 2 } - 5 t + 3$$

1. Find the times at which the particle is instantaneously at rest. Hence or otherwise find the minimum velocity of the particle.
2. Sketch the velocity-time graph for the first 3 seconds of motion.
3. Find the distance travelled between the two times when the particle is instantaneously at rest.

3 \includegraphics[max width=\textwidth, alt={}, center]{4e555003-16f1-4453-ab25-c50929d4b5b3-04_824_1636_264_258} The displacement of a particle moving in a straight line is \(s\) metres at time \(t\) seconds after leaving a fixed point \(O\). The particle starts from rest and passes through points \(P , Q\) and \(R\), at times \(t = 5 , t = 10\) and \(t = 15\) respectively, and returns to \(O\) at time \(t = 20\). The distances \(O P , O Q\) and \(O R\) are 50 m , 150 m and 200 m respectively. The diagram shows a displacement-time graph which models the motion of the particle from \(t = 0\) to \(t = 20\). The graph consists of two curved segments \(A B\) and \(C D\) and two straight line segments \(B C\) and \(D E\).

1. Find the speed of the particle between \(t = 5\) and \(t = 10\).
2. Find the acceleration of the particle between \(t = 0\) and \(t = 5\), given that it is constant.
3. Find the average speed of the particle during its motion. \includegraphics[max width=\textwidth, alt={}, center]{4e555003-16f1-4453-ab25-c50929d4b5b3-06_483_880_258_630} The diagram shows a block of mass 10 kg suspended below a horizontal ceiling by two strings \(A C\) and \(B C\), of lengths 0.8 m and 0.6 m respectively, attached to fixed points on the ceiling. Angle \(A C B = 90 ^ { \circ }\). There is a horizontal force of magnitude \(F \mathrm {~N}\) acting on the block. The block is in equilibrium.

7 A particle \(P\) moves in a straight line through a point \(O\). The velocity \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) of \(P\), at time \(t \mathrm {~s}\) after passing \(O\), is given by $$v = \frac { 9 } { 4 } + \frac { b } { ( t + 1 ) ^ { 2 } } - c t ^ { 2 }$$ where \(b\) and \(c\) are positive constants. At \(t = 5\), the velocity of \(P\) is zero and its acceleration is \(- \frac { 13 } { 12 } \mathrm {~m} \mathrm {~s} ^ { - 2 }\).

1. Show that \(b = 9\) and find the value of \(c\).
2. Given that the velocity of \(P\) is zero only at \(t = 5\), find the distance travelled in the first 10 seconds of motion.
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

6 A particle \(P\) starts at rest and moves in a straight line from a point \(O\). At time \(t\) s after leaving \(O\), the velocity of \(P , v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), is given by \(v = b t + c t ^ { \frac { 3 } { 2 } }\), where \(b\) and \(c\) are constants. \(P\) has velocity \(8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) when \(t = 4\) and has velocity \(13.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) when \(t = 9\).

1. Show that \(b = 3\) and \(c = - 0.5\).
2. Find the acceleration of \(P\) when \(t = 1\).
3. Find the positive value of \(t\) when \(P\) is at instantaneous rest and find the distance of \(P\) from \(O\) at this instant.
4. Find the speed of \(P\) at the instant it returns to \(O\).

7 A particle moves in a straight line through the point \(O\). The displacement of the particle from \(O\) at time \(t \mathrm {~s}\) is \(s \mathrm {~m}\), where $$\begin{array} { l l } s = t ^ { 2 } - 3 t + 2 & \text { for } 0 \leqslant t \leqslant 6 , \\ s = \frac { 24 } { t } - \frac { t ^ { 2 } } { 4 } + 25 & \text { for } t \geqslant 6 . \end{array}$$

1. Find the value of \(t\) when the particle is instantaneously at rest during the first 6 seconds of its motion.
   At \(t = 6\), the particle hits a barrier at a point \(P\) and rebounds.
2. Find the velocity with which the particle arrives at \(P\) and also the velocity with which the particle leaves \(P\).
3. Find the total distance travelled by the particle in the first 10 seconds of its motion.
   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 A particle moves in a straight line. Its displacement \(t\) seconds after leaving the fixed point \(O\) is \(x\) metres, where \(x = \frac { 1 } { 2 } t ^ { 2 } + \frac { 1 } { 30 } t ^ { 3 }\). Find

1. the speed of the particle when \(t = 10\),
2. the value of \(t\) for which the acceleration of the particle is twice its initial acceleration.

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 motorcyclist starts from rest at \(A\) and travels in a straight line until he comes to rest again at \(B\). The velocity of the motorcyclist \(t\) seconds after leaving \(A\) is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), where \(v = t - 0.01 t ^ { 2 }\). Find

1. the time taken for the motorcyclist to travel from \(A\) to \(B\),
2. the distance \(A B\).

6 A particle starts from rest at a point \(O\) and moves in a horizontal straight line. The velocity of the particle is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at time \(t \mathrm {~s}\) after leaving \(O\). For \(0 \leqslant t < 60\), the velocity is given by $$v = 0.05 t - 0.0005 t ^ { 2 }$$ The particle hits a wall at the instant when \(t = 60\), and reverses the direction of its motion. The particle subsequently comes to rest at the point \(A\) when \(t = 100\), and for \(60 < t \leqslant 100\) the velocity is given by $$v = 0.025 t - 2.5$$

1. Find the velocity of the particle immediately before it hits the wall, and its velocity immediately after its hits the wall.
2. Find the total distance travelled by the particle.
3. Find the maximum speed of the particle and sketch the particle's velocity-time graph for \(0 \leqslant t \leqslant 100\), showing the value of \(t\) for which the speed is greatest. \section*{[Question 7 is printed on the next page.]}

6 A particle \(P\) moves in a straight line. It starts at a point \(O\) on the line and at time \(t\) s after leaving \(O\) it has a velocity \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), where \(v = 6 t ^ { 2 } - 30 t + 24\).

1. Find the set of values of \(t\) for which the acceleration of the particle is negative.
2. Find the distance between the two positions at which \(P\) is at instantaneous rest.
3. Find the two positive values of \(t\) at which \(P\) passes through \(O\).

2 A particle \(P\) moves in a straight line, starting from a point \(O\). At time \(t \mathrm {~s}\) after leaving \(O\), the velocity of \(P , v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), is given by \(v = 4 t ^ { 2 } - 8 t + 3\).

1. Find the two values of \(t\) at which \(P\) is at instantaneous rest.
2. Find the distance travelled by \(P\) between these two times.

7 A particle \(P\) moves in a straight line. At time \(t \mathrm {~s}\), the displacement of \(P\) from \(O\) is \(s \mathrm {~m}\) and the acceleration of \(P\) is \(a \mathrm {~m} \mathrm {~s} ^ { - 2 }\), where \(a = 6 t - 2\). When \(t = 1 , s = 7\) and when \(t = 3 , s = 29\).

1. Find the set of values of \(t\) for which the particle is decelerating.
2. Find \(s\) in terms of \(t\).
3. Find the time when the velocity of the particle is \(10 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

6 A particle \(P\) moves in a straight line passing through a point \(O\). At time \(t \mathrm {~s}\), the velocity of \(P , v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), is given by \(v = q t + r t ^ { 2 }\), where \(q\) and \(r\) are constants. The particle has velocity \(4 \mathrm {~ms} ^ { - 1 }\) when \(t = 1\) and when \(t = 2\).

1. Show that, when \(t = 0.5\), the acceleration of \(P\) is \(4 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).
   ............................................................................................................................... .
2. Find the values of \(t\) when \(P\) is at instantaneous rest.
3. The particle is at \(O\) when \(t = 3\). Find the distance of \(P\) from \(O\) when \(t = 0\).

2 A particle \(P\) of mass \(m \mathrm {~kg}\) moves on an arc of a circle with centre \(O\) and radius \(a\) metres. At time \(t = 0\) the particle is at the point \(A\). At time \(t\) seconds, angle \(P O A = \sin ^ { 2 } 2 t\). Show that the radial component of the acceleration of \(P\) at time \(t\) seconds has magnitude \(\left( 4 a \sin ^ { 2 } 4 t \right) \mathrm { m } \mathrm { s } ^ { - 2 }\). Find

1. the value of \(t\) when the transverse component of the acceleration of \(P\) is first equal to zero,
2. the magnitude of the resultant force acting on \(P\) when \(t = \frac { 1 } { 12 } \pi\).

4 A particle \(P\) moves in a straight line starting from a point \(O\). At time \(t \mathrm {~s}\) after leaving \(O\), the velocity, \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), of \(P\) is given by \(v = ( 2 t - 5 ) ^ { 3 }\).

1. Find the values of \(t\) when the acceleration of \(P\) is \(54 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).
2. Find an expression for the displacement of \(P\) from \(O\) at time \(t \mathrm {~s}\).

5 \includegraphics[max width=\textwidth, alt={}, center]{2026cad4-8494-4139-ad21-d8a17ac2b955-3_917_1451_1059_347} The diagram shows the displacement-time graph for a car's journey. The graph consists of two curved parts \(A B\) and \(C D\), and a straight line \(B C\). The line \(B C\) is a tangent to the curve \(A B\) at \(B\) and a tangent to the curve \(C D\) at \(C\). The gradient of the curves at \(t = 0\) and \(t = 600\) is zero, and the acceleration of the car is constant for \(0 < t < 80\) and for \(560 < t < 600\). The displacement of the car is 400 m when \(t = 80\).

1. Sketch the velocity-time graph for the journey.
2. Find the velocity at \(t = 80\).
3. Find the total distance for the journey.
4. Find the acceleration of the car for \(0 < t < 80\).

2 \includegraphics[max width=\textwidth, alt={}, center]{9c7e8624-c4cd-4a8e-83d9-f92d0bd6f95b-2_262_1004_760_575} The tops of each of two smooth inclined planes \(A\) and \(B\) meet at a right angle. Plane \(A\) is inclined at angle \(\alpha\) to the horizontal and plane \(B\) is inclined at angle \(\beta\) to the horizontal, where \(\sin \alpha = \frac { 63 } { 65 }\) and \(\sin \beta = \frac { 16 } { 65 }\). A small smooth pulley is fixed at the top of the planes and a light inextensible string passes over the pulley. Two particles \(P\) and \(Q\), each of mass 0.65 kg , are attached to the string, one at each end. Particle \(Q\) is held at rest at a point of the same line of greatest slope of the plane \(B\) as the pulley. Particle \(P\) rests freely below the pulley in contact with plane \(A\) (see diagram). Particle \(Q\) is released and the particles start to move with the string taut. Find the tension in the string.

4 A particle \(P\) starts from rest and moves in a straight line for 18 seconds. For the first 8 seconds of the motion \(P\) has constant acceleration \(0.25 \mathrm {~m} \mathrm {~s} ^ { - 2 }\). Subsequently \(P\) 's velocity, \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at time \(t\) seconds after the motion started, is given by $$v = - 0.1 t ^ { 2 } + 2.4 t - k ,$$ where \(8 \leqslant t \leqslant 18\) and \(k\) is a constant.

1. Find the value of \(v\) when \(t = 8\) and hence find the value of \(k\).
2. Find the maximum velocity of \(P\).
3. Find the displacement of \(P\) from its initial position when \(t = 18\).

5 A box of mass 8 kg is on a rough plane inclined at \(5 ^ { \circ }\) to the horizontal. A force of magnitude \(P \mathrm {~N}\) acts on the box in a direction upwards and parallel to a line of greatest slope of the plane. When \(P = 7 X\) the box moves up the line of greatest slope with acceleration \(0.15 \mathrm {~ms} ^ { - 2 }\) and when \(P = 8 X\) the box moves up the line of greatest slope with acceleration \(1.15 \mathrm {~ms} ^ { - 2 }\). Find the value of \(X\) and the coefficient of friction between the box and the plane.

6 \begin{figure}[h] \end{figure} \begin{figure}[h] \end{figure} Particles \(P\) and \(Q\) have a total mass of 1 kg . The particles are attached to opposite ends of a light inextensible string which passes over a smooth fixed pulley. \(P\) is held at rest and \(Q\) hangs freely, with both straight parts of the string vertical. Both particles are at a height of \(h \mathrm {~m}\) above the floor (see Fig. 1). \(P\) is released from rest and the particles start to move with the string taut. Fig. 2 shows the velocity-time graphs for \(P\) 's motion and for \(Q\) 's motion, where the positive direction for velocity is vertically upwards. Find

1. the magnitude of the acceleration with which the particles start to move and the mass of each of the particles,
2. the value of \(h\),
3. the greatest height above the floor reached by particle \(P\).

7 \includegraphics[max width=\textwidth, alt={}, center]{9c7e8624-c4cd-4a8e-83d9-f92d0bd6f95b-4_668_848_260_653} A small block of mass 3 kg is initially at rest at the bottom \(O\) of a rough plane inclined at an angle \(\alpha\) to the horizontal, where \(\sin \alpha = 0.6\) and \(\cos \alpha = 0.8\). A force of magnitude 35 N acts on the block at an angle \(\beta\) above the plane, where \(\sin \beta = 0.28\) and \(\cos \beta = 0.96\). The block starts to move up a line of greatest slope of the plane and passes through a point \(A\) with speed \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The distance \(O A\) is 12.5 m (see diagram).

1. For the motion of the block from \(O\) to \(A\), find the work done against the frictional force acting on the block.
2. Find the coefficient of friction between the block and the plane. At the instant that the block passes through \(A\) the force of magnitude 35 N ceases to act.
3. Find the distance the block travels up the plane after passing through \(A\). \end{document}

3 A particle \(P\) moves along a straight line for 100 s . It starts at a point \(O\) and at time \(t\) seconds after leaving \(O\) the velocity of \(P\) is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), where $$v = 0.00004 t ^ { 3 } - 0.006 t ^ { 2 } + 0.288 t$$

1. Find the values of \(t\) at which the acceleration of \(P\) is zero.
2. Find the displacement of \(P\) from \(O\) when \(t = 100\).

6 A particle \(P\) starts from rest at a point \(O\) of a straight line and moves along the line. The displacement of the particle at time \(t \mathrm {~s}\) after leaving \(O\) is \(x \mathrm {~m}\), where $$x = 0.08 t ^ { 2 } - 0.0002 t ^ { 3 }$$

1. Find the value of \(t\) when \(P\) returns to \(O\) and find the speed of \(P\) as it passes through \(O\) on its return.
2. For the motion of \(P\) until the instant it returns to \(O\), find
   1. the total distance travelled,
   2. the average speed.

6 A particle \(P\) of mass 0.2 kg is projected horizontally from a fixed point \(O\) on a smooth horizontal surface. When the displacement of \(P\) from \(O\) is \(x \mathrm {~m}\) the velocity of \(P\) is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\). A horizontal force of variable magnitude \(0.09 \sqrt { } x \mathrm {~N}\) directed away from \(O\) acts on \(P\). An additional force of constant magnitude 0.3 N directed towards \(O\) acts on \(P\).

1. Show that \(v \frac { \mathrm {~d} v } { \mathrm {~d} x } = 0.45 \sqrt { } x - 1.5\).
2. Find the value of \(x\) for which the acceleration of \(P\) is zero.
3. Given that the minimum value of \(v\) is positive, find the set of possible values for the speed of projection.