Questions M1 (2067 questions)

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OCR MEI M1 Q4
8 marks Moderate -0.8
4 You should neglect air resistance in this question.
A small stone is projected from ground level. The maximum height of the stone above horizontal ground is 22.5 m .
  1. Show that the vertical component of the initial velocity of the stone is \(21 \mathrm {~ms} { } ^ { 1 }\). The speed of projection is \(28 \mathrm {~ms} { } ^ { 1 }\).
  2. Find the angle of projection of the stone.
  3. Find the horizontal range of the stone.
OCR MEI M1 Q5
19 marks Standard +0.3
5 In this question take the value of \(\boldsymbol { g }\) to be \(\mathbf { 1 0 ~ } \mathbf { m ~ s } ^ { \mathbf { 2 } }\). \(\Lambda\) particle \(\Lambda\) is projected over horizontal ground from a point P which is 9 m above a point O on the ground. The initial velocity has horizontal and vertical components of \(10 \mathrm {~ms} ^ { - 1 }\) and \(12 \mathrm {~ms} ^ { - 1 }\) respectively, as shown in Fig. 7. The trajectory of the particle meets the ground at X. Air resistance may be neglected. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{9eab8ba4-d97b-4e3a-b36d-53f4bc7a80c2-3_394_788_551_630} \captionsetup{labelformat=empty} \caption{Fig. 7}
\end{figure}
  1. Calculate the specd of projection \(u \mathrm {~ms} ^ { - 1 }\) and the angle of projection \(\theta ^ { \circ }\).
  2. Show that, \(t\) seconds after projection, the height of particle A above the ground is \(9 + 12 t - 5 t ^ { 2 }\). Write down an expression in terms of \(t\) for the horizontal distance of the particle from O at this time.
  3. Calculate the maximum height of particle \(\Lambda\) above the point of projection.
  4. Calculate the distance OX . \(\wedge\) second particle, \(B\), is projected from \(O\) with speed \(20 \mathrm {~ms} ^ { - 1 }\) at \(60 ^ { \circ }\) to the horizontal. The trajectories of A and B are in the same vertical plane. Particles A and B are projected at the same time.
  5. Show that the horizontal displacements of A and B are always cqual.
  6. Show that, \(t\) seconds after projection, the height of particle B above the ground is \(10 \sqrt { 3 } t - 5 t ^ { 2 }\).
  7. Show that the particles collide 1.7 seconds after projection (correct to two significant figures).
OCR MEI M1 Q6
7 marks Moderate -0.8
6 Ali is throwing flat stones onto water, hoping that they will bounce, as illustrated in Fig. 5.
Ali throws one stone from a height of 1.225 m above the water with initial speed \(20 \mathrm {~ms} ^ { - 1 }\) in a horizontal direction. Air resistance should be neglected. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{9eab8ba4-d97b-4e3a-b36d-53f4bc7a80c2-4_233_959_482_575} \captionsetup{labelformat=empty} \caption{Fig. 5}
\end{figure}
  1. Find the time it takes for the stone to reach the water.
  2. Find the speed of the stone when it reaches the water and the angle its trajectory makes with the horizontal at this time.
OCR MEI M1 Q7
8 marks Moderate -0.8
7 A projectile P travels in a vertical plane over level ground. Its position vector \(\mathbf { r }\) at time \(t\) seconds after projection is modelled by $$\mathbf { r } = \binom { x } { y } = \binom { 0 } { 5 } + \binom { 30 } { 40 } t - \binom { 0 } { 5 } t ^ { 2 }$$ where distances are in metres and the origin is a point on the level ground.
  1. Write down
    (A) the height from which P is projected,
    (B) the value of \(g\) in this model.
  2. Find the displacement of P from \(t = 3\) to \(t = 5\).
  3. Show that the equation of the trajectory is $$y = 5 + \frac { 4 } { 3 } x - \frac { x ^ { 2 } } { 180 }$$
OCR MEI M1 Q1
19 marks Standard +0.3
1 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. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{362d5995-bd39-4b07-b6a4-63eb1dd3e69d-1_611_1047_486_538} \captionsetup{labelformat=empty} \caption{Fig. 7}
\end{figure} The initial horizontal component of the velocity of the firework is \(21 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
  1. Calculate the time for the firework to reach its highest point and show that the initial vertical component of its velocity is \(28 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
  2. Show that the firework is \(\left( 28 t - 4.9 t ^ { 2 } \right) \mathrm { m }\) above the ground \(t\) seconds after its projection. 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 \(21 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and the other with a quarter of this speed.
  3. 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.
  4. Find the distance between these parts of the firework
    (A) when they reach the ground,
    (B) when they are 10 m above the ground.
  5. Show that the cartesian equation of the trajectory of the firework before it explodes is \(y = \frac { 1 } { 90 } \left( 120 x - x ^ { 2 } \right)\), referred to the coordinate axes shown in Fig. 7.
OCR MEI M1 Q1
20 marks Standard +0.3
1 A girl throws a small stone with initial speed \(14 \mathrm {~ms} { } ^ { 1 }\) at an angle of \(60 ^ { \circ }\) to the horizontal from a point 1 m above the ground. She throws the stone directly towards a vertical wall of height 6 m standing on horizontal ground. The point O is on the ground directly below the point of projection, as shown in Fig. 8. Air resistance is negligible. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{4e0ddc86-c340-4057-bf3a-1c98587c3110-1_666_757_416_679} \captionsetup{labelformat=empty} \caption{Fig. 8}
\end{figure}
  1. Write down an expression in terms of \(t\) for the horizontal displacement of the stone from O , \(t\) seconds after projection. Find also an expression for the height of the stone above O at this time. The stone is at the top of its trajectory when it passes over the wall.
  2. (A) Find the time it takes for the stone to reach its highest point.
    (B) Calculate the distance of O from the base of the wall.
    (C) Show that the stone passes over the wall with 2.5 m clearance.
  3. Find the cartesian equation of the trajectory of the stone referred to the horizontal and vertical axes, \(\mathrm { O } x\) and \(\mathrm { O } y\). There is no need to simplify your answer. The girl now moves away a further distance \(d \mathrm {~m}\) from the wall. She throws a stone as before and it just passes over the wall.
  4. Calculate \(d\).
OCR MEI M1 Q2
8 marks Standard +0.3
2 A particle is projected vertically upwards from a point O at \(21 \mathrm {~ms} ^ { - 1 }\).
  1. Calculate the greatest height reached by the particle. When this particle is at its highest point, a second particle is projected vertically upwards from \(O\) at \(15 \mathrm {~ms} ^ { - 1 }\).
  2. Show that the particles collide 1.5 seconds later and determine the height above O at which the collision takes place.
OCR MEI M1 Q3
17 marks Standard +0.3
3 The trajectory ABCD of a small stone moving with negligible air resistance is shown in Fig. 7. AD is horizontal and BC is parallel to AD . The stone is projected from A with speed \(40 \mathrm {~ms} ^ { - 1 }\) at \(50 ^ { \circ }\) to the horizontal. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{4e0ddc86-c340-4057-bf3a-1c98587c3110-3_316_1032_583_504} \captionsetup{labelformat=empty} \caption{Fig. 7}
\end{figure}
  1. Write down an expression for the horizontal displacement from A of the stone \(t\) seconds after projection. Write down also an expression for the vertical displacement at time \(t\).
  2. Show that the stone takes 6.253 seconds (to three decimal places) to travel from A to D . Calculate the range of the stone. You are given that \(X = 30\).
  3. Calculate the time it takes the stone to reach B . Hence determine the time for it to travel from A to C.
  4. Calculate the direction of the motion of the stone at C .
OCR MEI M1 Q1
3 marks Easy -1.3
1 The vectors \(\mathbf { P } , \mathbf { Q }\) and \(\mathbf { R }\) are given by $$\mathbf { P } = 5 \mathbf { i } + 4 \mathbf { j } , \quad \mathbf { Q } = 3 \mathbf { i } - 5 \mathbf { j } , \quad \mathbf { R } = - 8 \mathbf { i } + \mathbf { j }$$
  1. Find the vector \(\mathbf { P } + \mathbf { Q } + \mathbf { R }\).
  2. Interpret your answer to part (i) in the cases
    (A) \(\mathbf { P } , \mathbf { Q }\) and \(\mathbf { R }\) represent three forces acting on a particle,
    (B) \(\mathbf { P } , \mathbf { Q }\) and \(\mathbf { R }\) represent three stages of a hiker's walk.
OCR MEI M1 Q2
3 marks Easy -1.3
2 The vectors \(\mathbf { P } , \mathbf { Q }\) and \(\mathbf { R }\) are given by $$\mathbf { P } = 5 \mathbf { i } + 4 \mathbf { j } , \quad \mathbf { Q } = 3 \mathbf { i } - 5 \mathbf { j } , \quad \mathbf { R } = - 8 \mathbf { i } + \mathbf { j }$$
  1. Find the vector \(\mathbf { P } + \mathbf { Q } + \mathbf { R }\).
  2. Interpret your answer to part (i) in the cases
    (A) \(\mathbf { P } , \mathbf { Q }\) and \(\mathbf { R }\) represent three forces acting on a particle,
    (B) \(\mathbf { P } , \mathbf { Q }\) and \(\mathbf { R }\) represent three stages of a hiker's walk.
OCR MEI M1 Q3
6 marks Moderate -0.3
3 In this question the unit vectors \(\mathbf { i }\) and \(\mathbf { j }\) are pointing east and north respectively.
  1. Calculate the bearing of the vector \(- 4 \mathbf { i } - 6 \mathbf { j }\). The vector \(- 4 \mathbf { i } - 6 \mathbf { j } + k ( 3 \mathbf { i } - 2 \mathbf { j } )\) is in the direction \(7 \mathbf { i } - 9 \mathbf { j }\).
  2. Find \(k\).
OCR MEI M1 Q4
7 marks Moderate -0.8
4 A small box has weight \(\mathbf { W } \mathrm { N }\) and is held in equilibrium by two strings with tensions \(\mathbf { T } _ { 1 } \mathrm {~N}\) and \(\mathbf { T } _ { 2 } \mathrm {~N}\). This situation is shown in Fig. 2 which also shows the standard unit vectors \(\mathbf { i }\) and \(\mathbf { j }\) that are horizontal and vertically upwards, respectively. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{b80eced6-2fea-4b95-9104-d13339643df0-2_252_631_414_803} \captionsetup{labelformat=empty} \caption{Fig. 2}
\end{figure} The tension \(\mathbf { T } _ { 1 }\) is \(10 \mathbf { i } + 24 \mathbf { j }\).
  1. Calculate the magnitude of \(\mathbf { T } _ { 1 }\) and the angle between \(\mathbf { T } _ { 1 }\) and the vertical. The magnitude of the weight is \(w \mathrm {~N}\).
  2. Write down the vector \(\mathbf { W }\) in terms of \(w\) and \(\mathbf { j }\). The tension \(\mathbf { T } _ { 2 }\) is \(k \mathbf { i } + 10 \mathbf { j }\), where \(k\) is a scalar.
  3. Find the values of \(k\) and of \(w\).
OCR MEI M1 Q5
6 marks Moderate -0.8
5 A particle has a position vector \(\mathbf { r }\), where \(\mathbf { r } = 4 \mathbf { i } - 5 \mathbf { j }\) and \(\mathbf { i }\) and \(\mathbf { j }\) are unit vectors in the directions east and north respectively.
  1. Sketch \(\mathbf { r }\) on a diagram showing \(\mathbf { i }\) and \(\mathbf { j }\) and the origin O .
  2. Calculate the magnitude of \(\mathbf { r }\) and its direction as a bearing.
  3. Write down the vector that has the same direction as \(\mathbf { r }\) and three times its magnitude.
OCR MEI M1 Q6
8 marks Moderate -0.3
6 Force \(\mathbf { F } _ { 1 }\) is \(\binom { 6 } { 13 } \mathrm {~N}\) and force \(\mathbf { F } _ { 2 }\) is \(\binom { 3 } { 5 }\), where \(\left. \int _ { 0 } \right] _ { \text {and } } \binom { 0 } { 1 }\) are vectors east and north respectively.
  1. Calculate the magnitude of \(\mathbf { F } _ { 1 }\), correct to three significant figures.
  2. Calculate the direction of the force \(\mathbf { F } _ { 1 } - \mathbf { F } _ { 2 }\) as a bearing. Force \(\mathbf { F } _ { 2 }\) is the resultant of all the forces acting on an object of mass 5 kg .
  3. Calculate the acceleration of the object and the change in its velocity after 10 seconds.
OCR MEI M1 Q1
6 marks Moderate -0.8
1 A particle rests on a smooth, horizontal plane. Horizontal unit vectors \(\mathbf { i }\) and \(\mathbf { j }\) lie in this plane. The particle is in equilibrium under the action of the three forces \(( - 3 \mathbf { i } + 4 \mathbf { j } ) \mathrm { N }\) and \(( 21 \mathbf { i } - 7 \mathbf { j } ) \mathrm { N }\) and \(\mathbf { R N }\).
  1. Write down an expression for \(\mathbf { R }\) in terms of \(\mathbf { i }\) and \(\mathbf { j }\).
  2. Find the magnitude of \(\mathbf { R }\) and the angle between \(\mathbf { R }\) and the \(\mathbf { i }\) direction.
OCR MEI M1 Q2
7 marks Standard +0.3
2 The position vector of a particle at time \(t\) is given by $$\mathbf { r } = \frac { 1 } { 2 } t \mathbf { i } + \left( t ^ { 2 } - 1 \right) \mathbf { j } .$$ referred to an origin \(O\) where \(\mathbf { i }\) and \(\mathbf { j }\) are the standard unit vectors in the directions of the cartesian axes Ox and Oy respectively.
  1. Write down the value of \(t\) for which the \(x\)-coordinate of the position of the particle is 2 . Find the \(y\)-coordinate at this time.
  2. Show that the cartesian equation of the path of the particle is \(y = 4 x ^ { 2 } - 1\).
  3. Find the coordinates of the point where the particle is moving at \(45 ^ { \circ }\) to both Ox and Oy .
OCR MEI M1 Q3
8 marks Moderate -0.8
3 The vectors \(\mathbf { p }\) and \(\mathbf { q }\) are given by $$\mathbf { p } = 8 \mathbf { i } + \mathbf { j } \text { and } \mathbf { q } = 4 \mathbf { i } - 7 \mathbf { j } .$$
  1. Show that \(\mathbf { p }\) and \(\mathbf { q }\) are equal in magnitude.
  2. Show that \(\mathbf { p } + \mathbf { q }\) is parallel to \(2 \mathbf { i } - \mathbf { j }\).
  3. Draw \(\mathbf { p } + \mathbf { q }\) and \(\mathbf { p } - \mathbf { q }\) on the grid. Write down the angle between these two vectors.
OCR MEI M1 Q4
8 marks Moderate -0.3
4 In this question, \(\mathbf { i }\) is a horizontal unit vector and \(\mathbf { j }\) is a unit vector pointing vertically upwards.
A force \(\mathbf { F }\) is \(- \mathbf { i } + 5 \mathbf { j }\).
  1. Calculate the magnitude of \(\mathbf { F }\). Calculate also the angle between \(\mathbf { F }\) and the upward vertical. Force \(\mathbf { G }\) is \(2 a \mathbf { i } + a \mathbf { j }\) and force \(\mathbf { H }\) is \(- 2 \mathbf { i } + 3 b \mathbf { j }\), where \(a\) and \(b\) are constants. The force \(\mathbf { H }\) is the resultant of forces \(4 \mathbf { F }\) and \(\mathbf { G }\).
  2. Find \(\mathbf { G }\) and \(\mathbf { H }\).
OCR MEI M1 Q5
6 marks Moderate -0.3
5 The resultant of the force \(\binom { - 4 } { 8 } \mathrm {~N}\) and the force \(\mathbf { F }\) gives an object of mass 6 kg an acceleration of \(\binom { 2 } { 3 } \mathrm {~m} \mathrm {~s} ^ { - 2 }\).
  1. Calculate \(\mathbf { F }\).
  2. Calculate the angle between \(\mathbf { F }\) and the vector \(\binom { 0 } { 1 }\).
OCR MEI M1 Q6
7 marks Moderate -0.5
6 The force acting on a particle of mass 1.5 kg is given by the vector \(\binom { 6 } { 9 } \mathrm {~N}\).
  1. Give the acceleration of the particle as a vector.
  2. Calculate the angle that the acceleration makes with the direction \(\binom { 1 } { 0 }\).
  3. At a certain point of its motion, the particle has a velocity of \(\binom { - 2 } { 3 } \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Calculate the displacement of the particle over the next two seconds.
OCR MEI M1 Q7
7 marks Moderate -0.3
7 A force \(\mathbf { F }\) is given by \(\mathbf { F } = ( 3.5 \mathbf { i } + 12 \mathbf { j } ) \mathrm { N }\), where \(\mathbf { i }\) and \(\mathbf { j }\) are horizontal unit vectors east and north respectively.
  1. Calculate the magnitude of \(\mathbf { F }\) and also its direction as a bearing.
  2. \(\mathbf { G }\) is the force \(( 7 \mathbf { i } + 24 \mathbf { j } ) \mathrm { N }\). Show that \(\mathbf { G }\) and \(\mathbf { F }\) are in the same direction and compare their magnitudes.
  3. Force \(\mathbf { F } _ { 1 }\) is \(( 9 \mathbf { i } - 18 \mathbf { j } ) \mathrm { N }\) and force \(\mathbf { F } _ { 2 }\) is \(( 12 \mathbf { i } + q \mathbf { j } ) \mathrm { N }\). Find \(q\) so that the sum \(\mathbf { F } _ { 1 } + \mathbf { F } _ { 2 }\) is in the direction of \(\mathbf { F }\).
CAIE M1 2022 June Q4
8 marks Standard +0.3
  1. In the case where \(F = 20\), find the tensions in each of the strings.
  2. Find the greatest value of \(F\) for which the block remains in equilibrium in the position shown.
CAIE M1 2022 June Q6
10 marks Standard +0.3
  1. It is given that the plane \(B C\) is smooth and that the particles are released from rest. Find the tension in the string and the magnitude of the acceleration of the particles.
  2. It is given instead that the plane \(B C\) is rough. A force of magnitude 3 N is applied to \(Q\) directly up the plane along a line of greatest slope of the plane. Find the least value of the coefficient of friction between \(Q\) and the plane \(B C\) for which the particles remain at rest.
CAIE M1 2011 June Q4
7 marks Standard +0.3
  1. Make a rough copy of the diagram and shade the region whose area represents the displacement of \(P\) from \(X\) at the instant when \(Q\) starts. It is given that \(P\) has travelled 70 m at the instant when \(Q\) starts.
  2. Find the value of \(T\).
  3. Find the distance between \(P\) and \(Q\) when \(Q\) 's speed reaches \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
  4. Sketch a single diagram showing the displacement-time graphs for both \(P\) and \(Q\), with values shown on the \(t\)-axis at which the speed of either particle changes.
CAIE M1 2015 June Q6
11 marks Challenging +1.2
  1. Find the value of \(h\).
  2. Find the value of \(m\), and find also the tension in the string while \(Q\) is moving.
  3. The string is slack while \(Q\) is at rest on the ground. Find the total time from the instant that \(P\) is released until the string becomes taut again.