Questions M1 (2067 questions)

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OCR MEI M1 Q3
8 marks Moderate -0.8
In this question, the unit vectors \(\begin{pmatrix} 1 \\ 0 \end{pmatrix}\) and \(\begin{pmatrix} 0 \\ 1 \end{pmatrix}\) are in the directions east and north. Distance is measured in metres and time, \(t\), in seconds. A radio-controlled toy car moves on a flat horizontal surface. A child is standing at the origin and controlling the car. When \(t = 0\), the displacement of the car from the origin is \(\begin{pmatrix} 0 \\ -2 \end{pmatrix}\) m, and the car has velocity \(\begin{pmatrix} 2 \\ 0 \end{pmatrix}\) ms\(^{-1}\). The acceleration of the car is constant and is \(\begin{pmatrix} -1 \\ 1 \end{pmatrix}\) ms\(^{-2}\).
  1. Find the velocity of the car at time \(t\) and its speed when \(t = 8\). [4]
  2. Find the distance of the car from the child when \(t = 8\). [4]
OCR MEI M1 Q4
8 marks Moderate -0.3
At time \(t\) seconds, a particle has position with respect to an origin O given by the vector $$\mathbf{r} = \begin{pmatrix} 8t \\ 10t^2 - 2t^3 \end{pmatrix},$$ where \(\begin{pmatrix} 1 \\ 0 \end{pmatrix}\) and \(\begin{pmatrix} 0 \\ 1 \end{pmatrix}\) are perpendicular unit vectors east and north respectively and distances are in metres.
  1. When \(t = 1\), the particle is at P. Find the bearing of P from O. [2]
  2. Find the velocity of the particle at time \(t\) and show that it is never zero. [3]
  3. Determine the time(s), if any, when the acceleration of the particle is zero. [3]
OCR MEI M1 Q5
7 marks Moderate -0.8
A particle of mass 5 kg has constant acceleration. Initially, the particle is at \(\begin{pmatrix} -1 \\ 2 \end{pmatrix}\) m with velocity \(\begin{pmatrix} 2 \\ -3 \end{pmatrix}\) ms\(^{-1}\); after 4 seconds the particle has velocity \(\begin{pmatrix} 12 \\ 9 \end{pmatrix}\) ms\(^{-1}\).
  1. Calculate the acceleration of the particle. [2]
  2. Calculate the position of the particle at the end of the 4 seconds. [3]
  3. Calculate the force acting on the particle. [2]
OCR MEI M1 Q6
19 marks Moderate -0.3
A toy boat moves in a horizontal plane with position vector \(\mathbf{r} = x\mathbf{i} + y\mathbf{j}\), where \(\mathbf{i}\) and \(\mathbf{j}\) are the standard unit vectors east and north respectively. The origin of the position vectors is at O. The displacements \(x\) and \(y\) are in metres. First consider only the motion of the boat parallel to the \(x\)-axis. For this motion $$x = 8t - 2t^2.$$ The velocity of the boat in the \(x\)-direction is \(v_x\) ms\(^{-1}\).
  1. Find an expression in terms of \(t\) for \(v_x\) and determine when the boat instantaneously has zero speed in the \(x\)-direction. [3]
Now consider only the motion of the boat parallel to the \(y\)-axis. For this motion $$v_y = (t - 2)(3t - 2),$$ where \(v_y\) ms\(^{-1}\) is the velocity of the boat in the \(y\)-direction at time \(t\) seconds.
  1. Given that \(y = 3\) when \(t = 1\), use integration to show that \(y = t^3 - 4t^2 + 4t + 2\). [4]
The position vector of the boat is given in terms of \(t\) by \(\mathbf{r} = (8t - 2t^2)\mathbf{i} + (t^3 - 4t^2 + 4t + 2)\mathbf{j}\).
  1. Find the time(s) when the boat is due north of O and also the distance of the boat from O at any such times. [4]
  2. Find the time(s) when the boat is instantaneously at rest. Find the distance of the boat from O at any such times. [5]
  3. Plot a graph of the path of the boat for \(0 \leq t \leq 2\). [3]
OCR MEI M1 Q1
6 marks Easy -1.3
A cyclist starts from rest and takes 10 seconds to accelerate at a constant rate up to a speed of \(15\text{ m s}^{-1}\). After travelling at this speed for 20 seconds, the cyclist then decelerates to rest at a constant rate over the next 5 seconds.
  1. Sketch a velocity-time graph for the motion. [3]
  2. Calculate the distance travelled by the cyclist. [3]
OCR MEI M1 Q2
4 marks Moderate -0.8
Fig. 1 is the velocity-time graph for the motion of a body. The velocity of the body is \(v\text{ m s}^{-1}\) at time \(t\) seconds. \includegraphics{figure_1} The displacement of the body from \(t = 0\) to \(t = 100\) is 1400 m. Find the value of \(V\). [4]
OCR MEI M1 Q3
6 marks Moderate -0.8
A particle travels in a straight line during the time interval \(0 \leqslant t \leqslant 12\), where \(t\) is the time in seconds. Fig. 1 is the velocity-time graph for the motion. \includegraphics{figure_3}
  1. Calculate the acceleration of the particle in the interval \(0 < t < 6\). [2]
  2. Calculate the distance travelled by the particle from \(t = 0\) to \(t = 4\). [2]
  3. When \(t = 0\) the particle is at A. Calculate how close the particle gets to A during the interval \(4 \leqslant t \leqslant 12\). [2]
OCR MEI M1 Q4
19 marks Moderate -0.3
In this question take \(g\) as \(10\text{ m s}^{-2}\). A small ball is released from rest. It falls for 2 seconds and is then brought to rest over the next 5 seconds. This motion is modelled in the speed-time graph Fig. 6. \includegraphics{figure_4} For this model,
  1. calculate the distance fallen from \(t = 0\) to \(t = 7\), [3]
  2. find the acceleration of the ball from \(t = 2\) to \(t = 6\), specifying the direction, [3]
  3. obtain an expression in terms of \(t\) for the downward speed of the ball from \(t = 2\) to \(t = 6\), [3]
  4. state the assumption that has been made about the resistance to motion from \(t = 0\) to \(t = 2\). [1]
The part of the motion from \(t = 2\) to \(t = 7\) is now modelled by \(v = -\frac{3}{2}t^2 + \frac{19}{2}t + 7\).
  1. Verify that \(v\) agrees with the values given in Fig. 6 at \(t = 2\), \(t = 6\) and \(t = 7\). [2]
  2. Calculate the distance fallen from \(t = 2\) to \(t = 7\) according to this model. [7]
OCR MEI M1 Q5
18 marks Standard +0.3
A box of emergency supplies is dropped to victims of a natural disaster from a stationary helicopter at a height of 1000 metres. The initial velocity of the box is zero. At time \(t\) s after being dropped, the acceleration, \(a\text{ m s}^{-2}\), of the box in the vertically downwards direction is modelled by $$a = 10 - t \text{ for } 0 \leqslant t \leqslant 10,$$ $$a = 0 \text{ for } t > 10.$$
  1. Find an expression for the velocity, \(v\text{ m s}^{-1}\), of the box in the vertically downwards direction in terms of \(t\) for \(0 \leqslant t \leqslant 10\). Show that for \(t > 10\), \(v = 50\). [4]
  2. Draw a sketch graph of \(v\) against \(t\) for \(0 \leqslant t \leqslant 20\). [3]
  3. Show that the height, \(h\) m, of the box above the ground at time \(t\) s is given, for \(0 \leqslant t \leqslant 10\), by $$h = 1000 - 5t^2 + \frac{1}{6}t^3.$$ Find the height of the box when \(t = 10\). [4]
  4. Find the value of \(t\) when the box hits the ground. [2]
  5. Some of the supplies in the box are damaged when the box hits the ground. So measures are considered to reduce the speed with which the box hits the ground the next time one is dropped. Two different proposals are made. Carry out suitable calculations and then comment on each of them.
    1. The box should be dropped from a height of 500 m instead of 1000 m. [2]
    2. The box should be fitted with a parachute so that its acceleration is given by $$a = 10 - 2t \text{ for } 0 \leqslant t \leqslant 5,$$ $$a = 0 \text{ for } t > 5.$$ [3]
OCR MEI M1 Q6
7 marks Moderate -0.8
\includegraphics{figure_6} Particles P and Q move in the same straight line. Particle P starts from rest and has a constant acceleration towards Q of \(0.5\text{ m s}^{-2}\). Particle Q starts 125 m from P at the same time and has a constant speed of \(10\text{ m s}^{-1}\) away from P. The initial values are shown in Fig. 4.
  1. Write down expressions for the distances travelled by P and by Q at time \(t\) seconds after the start of the motion. [2]
  2. How much time does it take for P to catch up with Q and how far does P travel in this time? [5]
OCR MEI M1 Q1
18 marks Moderate -0.3
Fig. 7 shows the trajectory of an object which is projected from a point O on horizontal ground. Its initial velocity is \(40\text{ms}^{-1}\) at an angle of \(\alpha\) to the horizontal. \includegraphics{figure_1}
  1. Show that, according to the standard projectile model in which air resistance is neglected, the flight time, \(T\) s, and the range, \(R\) m, are given by $$T = \frac{80\sin\alpha}{g} \text{ and } R = \frac{3200\sin\alpha\cos\alpha}{g}.$$ [6] A company is designing a new type of ball and wants to model its flight.
  2. Initially the company uses the standard projectile model. Use this model to show that when \(\alpha = 30°\) and the initial speed is \(40\text{ms}^{-1}\), \(T\) is approximately \(4.08\) and \(R\) is approximately \(141.4\). Find the values of \(T\) and \(R\) when \(\alpha = 45°\). [3] The company tests the ball using a machine that projects it from ground level across horizontal ground. The speed of projection is set at \(40\text{ms}^{-1}\). When the angle of projection is set at \(30°\), the range is found to be \(125\) m.
  3. Comment briefly on the accuracy of the standard projectile model in this situation. [1] The company refines the model by assuming that the ball has a constant deceleration of \(2\text{ms}^{-2}\) in the horizontal direction. In this new model, the resistance to the vertical motion is still neglected and so the flight time is still \(4.08\) s when the angle of projection is \(30°\).
  4. Using the new model, with \(\alpha = 30°\), show that the horizontal displacement from the point of projection, \(x\) m at time \(t\) s, is given by $$x = 40t\cos 30° - t^2.$$ Find the range and hence show that this new model is reasonably accurate in this case. [4] The company then sets the angle of projection to \(45°\) while retaining a projection speed of \(40\text{ms}^{-1}\). With this setting the range of the ball is found to be \(135\) m.
  5. Investigate whether the new model is also accurate for this angle of projection. [3]
  6. Make one suggestion as to how the model could be further refined. [1]
OCR MEI M1 Q2
19 marks Moderate -0.3
\includegraphics{figure_2} Fig. 7 shows a platform \(10\) m long and \(2\) m high standing on horizontal ground. A small ball projected from the surface of the platform at one end, O, just misses the other end, P. The ball is projected at \(68.5°\) to the horizontal with a speed of \(U\text{ms}^{-1}\). Air resistance may be neglected. At time \(t\) seconds after projection, the horizontal and vertical displacements of the ball from O are \(x\) m and \(y\) m.
  1. Obtain expressions, in terms of \(U\) and \(t\), for
    1. \(x\),
    2. \(y\). [3]
  2. The ball takes \(T\) s to travel from O to P. Show that \(T = \frac{U\sin 68.5°}{4.9}\) and write down a second equation connecting \(U\) and \(T\). [4]
  3. Hence show that \(U = 12.0\) (correct to three significant figures). [3]
  4. Calculate the horizontal distance of the ball from the platform when the ball lands on the ground. [5]
  5. Use the expressions you found in part (i) to show that the cartesian equation of the trajectory of the ball in terms of \(U\) is $$y = x\tan 68.5° - \frac{4.9x^2}{U^2(\cos 68.5°)^2}.$$ Use this equation to show again that \(U = 12.0\) (correct to three significant figures). [4]
OCR MEI M1 Q3
18 marks Standard +0.3
\includegraphics{figure_3} Fig. 7 shows the graph of \(y = \frac{1}{100}(100 + 15x - x^2)\). For \(0 \leq x < 20\), this graph shows the trajectory of a small stone projected from the point Q where \(y\) m is the height of the stone above horizontal ground and \(x\) m is the horizontal displacement of the stone from O. The stone hits the ground at the point R.
  1. Write down the height of Q above the ground. [1]
  2. Find the horizontal distance from O of the highest point of the trajectory and show that this point is \(1.5625\) m above the ground. [5]
  3. Show that the time taken for the stone to fall from its highest point to the ground is \(0.565\) seconds, correct to 3 significant figures. [3]
  4. Show that the horizontal component of the velocity of the stone is \(22.1\text{ms}^{-1}\), correct to 3 significant figures. Deduce the time of flight from Q to R. [5]
  5. Calculate the speed at which the stone hits the ground. [4]
OCR MEI M1 Q4
6 marks Standard +0.3
Sandy is throwing a stone at a plum tree. The stone is thrown from a point O at a speed of \(35\text{ms}^{-1}\) at an angle of \(\alpha\) to the horizontal, where \(\cos\alpha = 0.96\). You are given that, \(t\) seconds after being thrown, the stone is \((9.8t - 4.9t^2)\) m higher than O. When descending, the stone hits a plum which is \(3.675\) m higher than O. Air resistance should be neglected. Calculate the horizontal distance of the plum from O. [6]
OCR MEI M1 Q5
7 marks Standard +0.3
Small stones A and B are initially in the positions shown in Fig. 6 with B a height \(H\) m directly above A. \includegraphics{figure_5} At the instant when B is released from rest, A is projected vertically upwards with a speed of \(29.4\text{ms}^{-1}\). Air resistance may be neglected. The stones collide \(T\) seconds after they begin to move. At this instant they have the same speed, \(V\text{ms}^{-1}\), and A is still rising. By considering when the speed of A upwards is the same as the speed of B downwards, or otherwise, show that \(T = 1.5\) and find the values of \(V\) and \(H\). [7]
CAIE M1 2014 June Q4
Standard +0.3
4 A particle \(P\) moves on a straight line, starting from rest at a point \(O\) of the line. The time after \(P\) starts to move is \(t \mathrm {~s}\), and the particle moves along the line with constant acceleration \(\frac { 1 } { 4 } \mathrm {~m} \mathrm {~s} ^ { - 2 }\) until it passes through a point \(A\) at time \(t = 8\). After passing through \(A\) the velocity of \(P\) is \(\frac { 1 } { 2 } t ^ { \frac { 2 } { 3 } } \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
  1. Find the acceleration of \(P\) immediately after it passes through \(A\). Hence show that the acceleration of \(P\) decreases by \(\frac { 1 } { 12 } \mathrm {~m} \mathrm {~s} ^ { - 2 }\) as it passes through \(A\).
  2. Find the distance moved by \(P\) from \(t = 0\) to \(t = 27\).
CAIE M1 2014 June Q5
Standard +0.3
5 \includegraphics[max width=\textwidth, alt={}, center]{77976dad-c055-45fd-93fe-e37fa8e9ae22-3_343_691_254_725} A light inextensible rope has a block \(A\) of mass 5 kg attached at one end, and a block \(B\) of mass 16 kg attached at the other end. The rope passes over a smooth pulley which is fixed at the top of a rough plane inclined at an angle of \(30 ^ { \circ }\) to the horizontal. Block \(A\) is held at rest at the bottom of the plane and block \(B\) hangs below the pulley (see diagram). The coefficient of friction between \(A\) and the plane is \(\frac { 1 } { \sqrt { 3 } }\). Block \(A\) is released from rest and the system starts to move. When each of the blocks has moved a distance of \(x \mathrm {~m}\) each has speed \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
  1. Write down the gain in kinetic energy of the system in terms of \(v\).
  2. Find, in terms of \(x\),
    (a) the loss of gravitational potential energy of the system,
    (b) the work done against the frictional force.
  3. Show that \(21 v ^ { 2 } = 220 x\).