Questions — OCR MEI (4455 questions)

Browse by module
All questions AEA AS Paper 1 AS Paper 2 AS Pure C1 C12 C2 C3 C34 C4 CP AS CP1 CP2 D1 D2 F1 F2 F3 FD1 FD1 AS FD2 FD2 AS FM1 FM1 AS FM2 FM2 AS FP1 FP1 AS FP2 FP2 AS FP3 FS1 FS1 AS FS2 FS2 AS Further AS Paper 1 Further AS Paper 2 Discrete Further AS Paper 2 Mechanics Further AS Paper 2 Statistics Further Additional Pure Further Additional Pure AS Further Discrete Further Discrete AS Further Extra Pure Further Mechanics Further Mechanics A AS Further Mechanics AS Further Mechanics B AS Further Mechanics Major Further Mechanics Minor Further Numerical Methods Further Paper 1 Further Paper 2 Further Paper 3 Further Paper 3 Discrete Further Paper 3 Mechanics Further Paper 3 Statistics Further Paper 4 Further Pure Core Further Pure Core 1 Further Pure Core 2 Further Pure Core AS Further Pure with Technology Further Statistics Further Statistics A AS Further Statistics AS Further Statistics B AS Further Statistics Major Further Statistics Minor Further Unit 1 Further Unit 2 Further Unit 3 Further Unit 4 Further Unit 5 Further Unit 6 H240/01 H240/02 H240/03 M1 M2 M3 M4 M5 P1 P2 P3 P4 PMT Mocks PURE Paper 1 Paper 2 Paper 3 Pre-U 9794/1 Pre-U 9794/2 Pre-U 9794/3 Pre-U 9795 Pre-U 9795/1 Pre-U 9795/2 S1 S2 S3 S4 Unit 1 Unit 2 Unit 3 Unit 4
Browse by board
AQA AS Paper 1 AS Paper 2 C1 C2 C3 C4 D1 D2 FP1 FP2 FP3 Further AS Paper 1 Further AS Paper 2 Discrete Further AS Paper 2 Mechanics Further AS Paper 2 Statistics Further Paper 1 Further Paper 2 Further Paper 3 Discrete Further Paper 3 Mechanics Further Paper 3 Statistics M1 M2 M3 Paper 1 Paper 2 Paper 3 S1 S2 S3 CAIE FP1 FP2 Further Paper 1 Further Paper 2 Further Paper 3 Further Paper 4 M1 M2 P1 P2 P3 S1 S2 Edexcel AEA AS Paper 1 AS Paper 2 C1 C12 C2 C3 C34 C4 CP AS CP1 CP2 D1 D2 F1 F2 F3 FD1 FD1 AS FD2 FD2 AS FM1 FM1 AS FM2 FM2 AS FP1 FP1 AS FP2 FP2 AS FP3 FS1 FS1 AS FS2 FS2 AS M1 M2 M3 M4 M5 P1 P2 P3 P4 PMT Mocks PURE Paper 1 Paper 2 Paper 3 S1 S2 S3 S4 OCR AS Pure C1 C2 C3 C4 D1 D2 FD1 AS FM1 AS FP1 FP1 AS FP2 FP3 FS1 AS Further Additional Pure Further Additional Pure AS Further Discrete Further Discrete AS Further Mechanics Further Mechanics AS Further Pure Core 1 Further Pure Core 2 Further Pure Core AS Further Statistics Further Statistics AS H240/01 H240/02 H240/03 M1 M2 M3 M4 PURE S1 S2 S3 S4 OCR MEI AS Paper 1 AS Paper 2 C1 C2 C3 C4 D1 D2 FP1 FP2 FP3 Further Extra Pure Further Mechanics A AS Further Mechanics B AS Further Mechanics Major Further Mechanics Minor Further Numerical Methods Further Pure Core Further Pure Core AS Further Pure with Technology Further Statistics A AS Further Statistics B AS Further Statistics Major Further Statistics Minor M1 M2 M3 M4 Paper 1 Paper 2 Paper 3 S1 S2 S3 S4 Pre-U Pre-U 9794/1 Pre-U 9794/2 Pre-U 9794/3 Pre-U 9795 Pre-U 9795/1 Pre-U 9795/2 WJEC Further Unit 1 Further Unit 2 Further Unit 3 Further Unit 4 Further Unit 5 Further Unit 6 Unit 1 Unit 2 Unit 3 Unit 4
Sort by: Default | Easiest first | Hardest first
OCR MEI M1 2008 January Q6
17 marks Moderate -0.3
A helicopter rescue activity at sea is modelled as follows. The helicopter is stationary and a man is suspended from it by means of a vertical, light, inextensible wire that may be raised or lowered, as shown in Fig. 6.1. \includegraphics{figure_6_1}
  1. When the man is descending with an acceleration 1.5 m s\(^{-2}\) downwards, how much time does it take for his speed to increase from 0.5 m s\(^{-1}\) downwards to 3.5 m s\(^{-1}\) downwards? How far does he descend in this time? [4]
The man has a mass of 80 kg. All resistances to motion may be neglected.
  1. Calculate the tension in the wire when the man is being lowered
    1. with an acceleration of 1.5 m s\(^{-2}\) downwards,
    2. with an acceleration of 1.5 m s\(^{-2}\) upwards. [5]
Subsequently, the man is raised and this situation is modelled with a constant resistance of 116 N to his upward motion.
  1. For safety reasons, the tension in the wire should not exceed 2500 N. What is the maximum acceleration allowed when the man is being raised? [4]
At another stage of the rescue, the man has equipment of mass 10 kg at the bottom of a vertical rope which is hanging from his waist, as shown in Fig. 6.2. The man and his equipment are being raised; the rope is light and inextensible and the tension in it is 80 N. \includegraphics{figure_6_2}
  1. Assuming that the resistance to the upward motion of the man is still 116 N and that there is negligible resistance to the motion of the equipment, calculate the tension in the wire. [4]
OCR MEI M1 2008 January Q7
19 marks Moderate -0.3
A small firework is fired from a point O at ground level over horizontal ground. The highest point reached by the firework is a horizontal distance of 60 m from O and a vertical distance of 40 m from O, as shown in Fig. 7. Air resistance is negligible.
[diagram]
The initial horizontal component of the velocity of the firework is 21 m s\(^{-1}\).
  1. Calculate the time for the firework to reach its highest point and show that the initial vertical component of its velocity is 28 m s\(^{-1}\). [4]
  2. Show that the firework is \((28t - 4.9t^2)\) m above the ground \(t\) seconds after its projection. [1]
When the firework is at its highest point it explodes into several parts. Two of the parts initially continue to travel horizontally in the original direction, one with the original horizontal speed of 21 m s\(^{-1}\) and the other with a quarter of this speed.
  1. State why the two parts are always at the same height as one another above the ground and hence find an expression in terms of \(t\) for the distance between the parts \(t\) seconds after the explosion. [3]
  2. Find the distance between these parts of the firework
    1. when they reach the ground, [2]
    2. when they are 10 m above the ground. [5]
  3. Show that the cartesian equation of the trajectory of the firework before it explodes is \(y = \frac{4}{90}(120x - x^2)\), referred to the coordinate axes shown in Fig. 7. [4]
OCR MEI M1 Q1
19 marks Moderate -0.3
The displacement, \(x\) m, from the origin O of a particle on the \(x\)-axis is given by $$x = 10 + 36t + 3t^2 - 2t^3,$$ where \(t\) is the time in seconds and \(-4 \leqslant t \leqslant 6\).
  1. Write down the displacement of the particle when \(t = 0\). [1]
  2. Find an expression in terms of \(t\) for the velocity, \(v\) ms\(^{-1}\), of the particle. [2]
  3. Find an expression in terms of \(t\) for the acceleration of the particle. [2]
  4. Find the maximum value of \(v\) in the interval \(-4 \leqslant t \leqslant 6\). [3]
  5. Show that \(v = 0\) only when \(t = -2\) and when \(t = 3\). Find the values of \(x\) at these times. [5]
  6. Calculate the distance travelled by the particle from \(t = 0\) to \(t = 4\). [3]
  7. Determine how many times the particle passes through O in the interval \(-4 \leqslant t \leqslant 6\). [3]
OCR MEI M1 Q2
8 marks Moderate -0.8
A particle moves along the \(x\)-axis with velocity, \(v\) ms\(^{-1}\), at time \(t\) given by $$v = 24t - 6t^2.$$ The positive direction is in the sense of \(x\) increasing.
  1. Find an expression for the acceleration of the particle at time \(t\). [2]
  2. Find the times, \(t_1\) and \(t_2\), at which the particle has zero speed. [2]
  3. Find the distance travelled between the times \(t_1\) and \(t_2\). [4]
OCR MEI M1 Q3
8 marks Moderate -0.3
Two girls, Marie and Nina, are members of an Olympic hockey team. They are doing fitness training. Marie runs along a straight line at a constant speed of \(6\) ms\(^{-1}\). Nina is stationary at a point O on the line until Marie passes her. Nina immediately runs after Marie until she catches up with her. The time, \(t\) s, is measured from the moment when Nina starts running. So when \(t = 0\), both girls are at O. Nina's acceleration, \(a\) ms\(^{-2}\), is given by \begin{align} a &= 4 - t \quad \text{for } 0 < t < 4,
a &= 0 \quad \text{for } t > 4. \end{align}
  1. Show that Nina's speed, \(v\) ms\(^{-1}\), is given by \begin{align} v &= 4t - \frac{1}{2}t^2 \quad \text{for } 0 < t < 4,
    v &= 8 \quad \text{for } t > 4. \end{align} [3]
  2. Find an expression for the distance Nina has run at time \(t\), for \(0 \leqslant t < 4\). Find how far Nina has run when \(t = 4\) and when \(t = 5\frac{1}{4}\). [4]
  3. Show that Nina catches up with Marie when \(t = 5\frac{1}{4}\). [1]
OCR MEI M1 Q4
7 marks Moderate -0.3
Two cars, P and Q, are being crashed as part of a film 'stunt'. At the start
  • P is travelling directly towards Q with a speed of \(8\) ms\(^{-1}\),
  • Q is instantaneously at rest and has an acceleration of \(4\) ms\(^{-2}\) directly towards P.
P continues with the same velocity and Q continues with the same acceleration. The cars collide \(T\) seconds after the start.
  1. Find expressions in terms of \(T\) for how far each of the cars has travelled since the start. [2]
At the start, P is 90 m from Q.
  1. Show that \(T^2 + 4T - 45 = 0\) and hence find \(T\). [5]
OCR MEI M1 Q5
8 marks Moderate -0.8
The velocity, \(v\) ms\(^{-1}\), of a particle moving along a straight line is given by $$v = 3t^2 - 12t + 14,$$ where \(t\) is the time in seconds.
  1. Find an expression for the acceleration of the particle at time \(t\). [2]
  2. Find the displacement of the particle from its position when \(t = 1\) to its position when \(t = 3\). [4]
  3. You are given that \(v\) is always positive. Explain how this tells you that the distance travelled by the particle between \(t = 1\) and \(t = 3\) has the same value as the displacement between these times. [2]
OCR MEI M1 Q1
8 marks Standard +0.3
A rock of mass 8 kg is acted on by just the two forces \(-80\)k N and \((-\mathbf{i} + 16\mathbf{j} + 72\)k\()\) N, where \(\mathbf{i}\) and \(\mathbf{j}\) are perpendicular unit vectors in a horizontal plane and k is a unit vector vertically upward.
  1. Show that the acceleration of the rock is \(\left(\frac{1}{8}\mathbf{i} + 2\mathbf{j}\right)\) k\()\) ms\(^{-2}\). [2]
The rock passes through the origin of position vectors, O, with velocity \((\mathbf{i} - 4\mathbf{j} + 3\)k\()\) m s\(^{-1}\) and 4 seconds later passes through the point A.
  1. Find the position vector of A. [3]
  2. Find the distance OA. [1]
  3. Find the angle that OA makes with the horizontal. [2]
OCR MEI M1 Q2
8 marks Moderate -0.3
Fig. 4 shows the unit vectors \(\mathbf{i}\) and \(\mathbf{j}\) in the directions of the cartesian axes \(Ox\) and \(Oy\), respectively. O is the origin of the axes and of position vectors. \includegraphics{figure_1} The position vector of a particle is given by \(\mathbf{r} = 3t\mathbf{i} + (18t^2 - 11)\mathbf{j}\) for \(t \geq 0\), where \(t\) is time.
  1. Show that the path of the particle cuts the \(x\)-axis just once. [2]
  2. Find an expression for the velocity of the particle at time \(t\). Deduce that the particle never travels in the \(\mathbf{j}\) direction. [3]
  3. Find the cartesian equation of the path of the particle, simplifying your answer. [3]
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]
OCR MEI M2 2007 January Q1
17 marks Moderate -0.3
A sledge and a child sitting on it have a combined mass of 29.5 kg. The sledge slides on horizontal ice with negligible resistance to its movement.
  1. While at rest, the sledge is hit directly from behind by a ball of mass 0.5 kg travelling horizontally at \(10 \text{ m s}^{-1}\). The coefficient of restitution in the collision is 0.8. After the impact the speeds of the sledge and the ball are \(V_1 \text{ m s}^{-1}\) and \(V_2 \text{ m s}^{-1}\) respectively. Calculate \(V_1\) and \(V_2\) and state the direction in which the ball is travelling after the impact. [7]
  2. While at rest, the sledge is hit directly from behind by a snowball of mass 0.5 kg travelling horizontally at \(10 \text{ m s}^{-1}\). The snowball sticks to the sledge.
    1. Calculate the velocity with which the combined sledge and snowball start to move. [3]
    2. The child scoops up the 0.5 kg of snow and drops it over the back of the sledge. What happens to the velocity of the sledge? Give a reason for your answer. [2]
  3. In another situation, the sledge is travelling over the ice at \(2 \text{ m s}^{-1}\) with 10.5 kg of snow on it (giving a total mass of 40 kg). The child throws a snowball of mass 0.5 kg from the sledge, parallel to the ground and in the positive direction of the motion of the sledge. Immediately after the snowball is thrown, the sledge has a speed of \(V \text{ m s}^{-1}\) and the snowball and sledge are separating at a speed of \(10 \text{ m s}^{-1}\). Draw a diagram showing the velocities of the sledge and snowball before and after the snowball is thrown. Calculate \(V\). [5]