Variable acceleration (vectors)

73 questions · 17 question types identified

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Find force using F=ma

A question is this type if and only if a velocity or position vector is given along with a mass, and the task requires finding the force vector (or its magnitude) by differentiating to get acceleration and applying Newton's second law.

14 Standard +0.1
19.2% of questions
3.02f Non-uniform acceleration: using differentiation and integration3.03d Newton's second law: 2D vectors
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2. A particle \(P\) of mass 0.5 kg moves under the action of a single force \(\mathbf { F }\) newtons. At time \(t\) seconds, the velocity \(\mathbf { v } \mathrm { m } \mathrm { s } ^ { - 1 }\) of \(P\) is given by $$\mathbf { v } = 3 t ^ { 2 } \mathbf { i } + ( 1 - 4 t ) \mathbf { j }$$ Find
  1. the acceleration of \(P\) at time \(t\) seconds,
  2. the magnitude of \(\mathbf { F }\) when \(t = 2\).
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Easiest question Moderate -0.8 »
1 The position vector, \(\mathbf { r }\), of a particle of mass 4 kg at time \(t\) is given by $$\mathbf { r } = t ^ { 2 } \mathbf { i } + \left( 5 t - 2 t ^ { 2 } \right) \mathbf { j } ,$$ where \(\mathbf { i }\) and \(\mathbf { j }\) are the standard unit vectors, lengths are in metres and time is in seconds.
  1. Find an expression for the acceleration of the particle. The particle is subject to a force \(\mathbf { F }\) and a force \(12 \mathbf { j } \mathbf { N }\).
  2. Find \(\mathbf { F }\).
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Hardest question Standard +0.3 »
3. A particle \(P\) of mass 0.25 kg is moving on a smooth horizontal surface under the action of a single force, \(\mathbf { F }\) newtons. At time \(t\) seconds \(( t \geqslant 0 )\), the velocity \(\mathbf { v } \mathrm { m } \mathrm { s } ^ { - 1 }\) of \(P\) is given by $$\mathbf { v } = ( 6 \sin 3 t ) \mathbf { i } + ( 1 + 2 \cos t ) \mathbf { j }$$
  1. Find \(\mathbf { F }\) in terms of \(t\). At time \(t = 0\), the position vector of \(P\) relative to a fixed point \(O\) is \(( 4 \mathbf { i } - \sqrt { 3 } \mathbf { j } ) \mathrm { m }\).
  2. Find the position vector of \(P\) relative to \(O\) when \(P\) is first moving parallel to the vector \(\mathbf { i }\).
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Find position by integrating velocity

A question is this type if and only if the velocity vector v(t) is given along with an initial or boundary condition for position, and the task requires integrating to find the position vector r(t) at a general or specific time.

10 Moderate -0.2
13.7% of questions
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A particle moves in a straight line with acceleration \(a\) m s\(^{-2}\), at time \(t\) seconds, where $$a = 10 - 6t$$ The particle's velocity, \(v\) m s\(^{-1}\), and displacement, \(r\) metres, are both initially zero. Show that $$r = t^2(5 - t)$$ Fully justify your answer. [4 marks]
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Easiest question Moderate -0.8 »
  1. \hspace{0pt} [In this question position vectors are given relative to a fixed origin \(O\) ]
At time \(t\) seconds, where \(t \geqslant 0\), a particle, \(P\), moves so that its velocity \(\mathbf { v ~ m ~ s } ^ { - 1 }\) is given by $$\mathbf { v } = 6 t \mathbf { i } - 5 t ^ { \frac { 3 } { 2 } } \mathbf { j }$$ When \(t = 0\), the position vector of \(P\) is \(( - 20 \mathbf { i } + 20 \mathbf { j } ) \mathrm { m }\).
  1. Find the acceleration of \(P\) when \(t = 4\)
  2. Find the position vector of \(P\) when \(t = 4\)
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Hardest question Standard +0.3 »
5. A t time \(t\) seconds ( \(t \geqslant 0\) ), a particle \(P\) has velocity \(\mathbf { v m ~ s } ^ { - 1 }\), where $$\mathbf { v } = \left( 3 t ^ { 2 } - 4 \right) \mathbf { i } + ( 2 t - 4 ) \mathbf { j }$$ When \(t = 0 , P\) is at the fixed point \(O\).
  1. Find the acceleration of \(P\) at the instant when \(t = 0\)
  2. Find the exact speed of \(P\) at the instant when \(P\) is moving in the direction of the vector \(( 11 \mathbf { i } + \mathbf { j } )\) for the second time.
  3. Show that \(P\) never returns to \(O\). \includegraphics[max width=\textwidth, alt={}, center]{c16c17b6-2c24-4939-b3b5-63cd63646b76-14_2658_1938_107_123} \includegraphics[max width=\textwidth, alt={}, center]{c16c17b6-2c24-4939-b3b5-63cd63646b76-15_149_140_2604_1818}
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Constant acceleration vector problems

A question is this type if and only if the acceleration is constant (given as a fixed vector or derived from a constant force), and the task involves using constant-acceleration vector equations to find velocity, position, or time.

10 Moderate -0.3
13.7% of questions
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A particle \(P\) moves with constant acceleration \((-4\mathbf{i} + 2\mathbf{j})\) ms\(^{-2}\). At time \(t = 0\) seconds, \(P\) is moving with velocity \((7\mathbf{i} + 6\mathbf{j})\) ms\(^{-1}\).
  1. Determine the speed of \(P\) when \(t = 3\). [4]
  2. Determine the change in displacement of \(P\) between \(t = 0\) and \(t = 3\). [2]
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Easiest question Moderate -0.8 »
2. A particle \(P\) of mass 1.5 kg is moving under the action of a constant force ( \(3 \mathbf { i } - 7.5 \mathbf { j }\) ) N. Initially \(P\) has velocity \(( 2 \mathbf { i } + 3 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\). Find
  1. the magnitude of the acceleration of \(P\),
  2. the velocity of \(P\), in terms of \(\mathbf { i }\) and \(\mathbf { j }\), when \(P\) has been moving for 4 seconds.
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Hardest question Standard +0.8 »
In this question the unit vectors \(\mathbf{i}\) and \(\mathbf{j}\) are in the directions east and north respectively. At time \(t\) seconds, where \(t \geqslant 0\), a particle \(P\) of mass 2 kg is moving on a smooth horizontal surface under the action of a constant horizontal force \((-8\mathbf{i} - 54\mathbf{j})\) N and a variable horizontal force \((4t\mathbf{i} + 6(2t - 1)^2\mathbf{j})\) N.
  1. Determine the value of \(t\) when the forces acting on \(P\) are in equilibrium. [2]
It is given that \(P\) is at rest when \(t = 0\).
  1. Determine the speed of \(P\) at the instant when \(P\) is moving due north. [6]
  2. Determine the distance between the positions of \(P\) when \(t = 0\) and \(t = 3\). [5]
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Find velocity from position

A question is this type if and only if the position vector r(t) is given and the task requires differentiating to find velocity (and possibly identifying when velocity has a specific direction or magnitude).

7 Moderate -0.0
9.6% of questions
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5 The position vector \(\mathbf { r }\) metres of a particle at time \(t\) seconds is given by $$\mathbf { r } = \left( 1 + 12 t - 2 t ^ { 2 } \right) \mathbf { i } + \left( t ^ { 2 } - 6 t \right) \mathbf { j }$$
  1. Find an expression for the velocity of the particle at time \(t\).
  2. Determine whether the particle is ever stationary.
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When moving parallel to given vector

A question is this type if and only if the task requires finding the time T (or value of a parameter) when the velocity vector is parallel to a specified vector or axis, by setting up a ratio or zero-component condition.

7 Standard +0.2
9.6% of questions
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3. At time \(t\) seconds, a particle \(P\) has position vector \(\mathbf { r }\) metres relative to a fixed origin \(O\), where $$\mathbf { r } = \left( t ^ { 3 } - 3 t \right) \mathbf { i } + 4 t ^ { 2 } \mathbf { j } , t \geq 0$$ Find
  1. the velocity of \(P\) at time \(t\) seconds,
  2. the time when \(P\) is moving parallel to the vector \(\mathbf { i } + \mathbf { j }\).
    (5)
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Cartesian equation of path

A question is this type if and only if the task requires eliminating the parameter t from the parametric position equations to produce a Cartesian equation y = f(x) for the path of the particle.

4 Moderate -0.3
5.5% of questions
1.10h Vectors in kinematics: uniform acceleration in vector form
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12 In this question the unit vectors \(\mathbf { i }\) and \(\mathbf { j }\) are in the \(x\) - and \(y\)-directions respectively.
The velocity \(\mathbf { v } \mathrm { m } \mathrm { s } ^ { - 1 }\) of a particle is given by \(\mathbf { v } = 3 \mathbf { i } + \left( 6 t ^ { 2 } - 5 \right) \mathbf { j }\). The initial position of the particle is \(7 \mathbf { j } \mathrm {~m}\).
  1. Find an expression for the position vector of the particle at time \(t \mathrm {~s}\).
  2. Find the Cartesian equation of the path of the particle.
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3D vector motion problems

A question is this type if and only if the motion involves three-dimensional vectors (using i, j, and k components), requiring analysis of position, velocity, acceleration, or force in three dimensions.

4 Standard +0.4
5.5% of questions
3.03d Newton's second law: 2D vectors
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2 A particle, Q , moves so that its velocity, \(\mathbf { v }\), at time \(t\) is given by \(\mathbf { v } = ( 6 t - 6 ) \mathbf { i } + \left( 3 - 2 t + t ^ { 2 } \right) \mathbf { j } + 4 \mathbf { k }\), where \(0 \leqslant t \leqslant 6\).
  1. Explain how you know that Q is never stationary. When Q is at a point A the direction of the acceleration of Q is parallel to the \(\mathbf { i }\) direction. When Q is at a point B the direction of the acceleration of Q makes an angle of \(45 ^ { \circ }\) with the \(\mathbf { i }\) direction.
  2. Determine the straight-line distance AB .
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Find velocity by integrating acceleration

A question is this type if and only if the acceleration vector a(t) is given along with an initial or boundary condition for velocity, and the task requires integrating to find the velocity vector v(t).

3 Moderate -0.1
4.1% of questions
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5 A particle moves such that at time \(t\) seconds its acceleration is given by $$( 2 \cos t \mathbf { i } - 5 \sin t \mathbf { j } ) \mathrm { m } \mathrm {~s} ^ { - 2 }$$
  1. The mass of the particle is 6 kg . Find the magnitude of the resultant force on the particle when \(t = 0\).
  2. When \(t = 0\), the velocity of the particle is \(( 2 \mathbf { i } + 10 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\). Find an expression for the velocity of the particle at time \(t\).
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Distance between two positions

A question is this type if and only if the task requires finding the scalar distance between two specific points (e.g. OA, AB, or OS) by computing position vectors at two times and finding the magnitude of their difference.

3 Standard +0.3
4.1% of questions
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4. A particle \(P\) of mass 0.4 kg is moving under the action of a single force \(\mathbf { F }\) newtons. At time \(t\) seconds, the velocity of \(P , \mathbf { v } \mathrm {~m} \mathrm {~s} ^ { - 1 }\), is given by $$\mathbf { v } = ( 6 t + 4 ) \mathbf { i } + \left( t ^ { 2 } + 3 t \right) \mathbf { j } .$$ When \(t = 0 , P\) is at the point with position vector \(( - 3 \mathbf { i } + 4 \mathbf { j } ) \mathrm { m }\). When \(t = 4 , P\) is at the point \(S\).
  1. Calculate the magnitude of \(\mathbf { F }\) when \(t = 4\).
  2. Calculate the distance \(O S\).
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Bearing or compass direction of motion

A question is this type if and only if the task requires finding the bearing (or compass direction such as north-east, south-east) on which a particle is travelling at a given time, using the components of the velocity vector.

2 Standard +0.3
2.7% of questions
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4 The directions of the unit vectors \(\mathbf { i }\) and \(\mathbf { j }\) are east and north.
The velocity of a particle, \(\mathrm { vm } \mathrm { s } ^ { - 1 }\), at time \(t \mathrm {~s}\) is given by $$\mathbf { v } = \left( 16 - t ^ { 2 } \right) \mathbf { i } + ( 31 - 8 t ) \mathbf { j } .$$ Find the time at which the particle is travelling on a bearing of \(045 ^ { \circ }\) and the speed of the particle at this time.
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Find unknown constant from motion condition

A question is this type if and only if the position or velocity expression contains an unknown constant (e.g. k or λ), and the task requires using a given condition on the direction or magnitude of motion to determine that constant.

2 Standard +0.3
2.7% of questions
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8 At time \(t\) seconds a particle \(P\) has position vector \(\mathbf { r }\) metres, with respect to a fixed origin \(O\), where $$\mathbf { r } = \left( 4 t ^ { 2 } - k t + 5 \right) \mathbf { i } + \left( 4 t ^ { 3 } + 2 k t ^ { 2 } - 8 t \right) \mathbf { j } , \quad t \geqslant 0 .$$ When \(t = 2 , P\) is moving parallel to the vector \(\mathbf { i }\).
  1. Show that \(k = - 5\).
  2. Find the values of \(t\) when the magnitude of the acceleration of \(P\) is \(10 \mathrm {~ms} ^ { - 2 }\).
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Magnitude of acceleration at given time

A question is this type if and only if the task specifically requires computing the magnitude (scalar value) of the acceleration vector at a particular time t, distinct from finding the force or the full acceleration vector.

2 Moderate -0.4
2.7% of questions
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A particle P has position vector \(\mathbf{r}\) m at time \(t\) s given by \(\mathbf{r} = (t^3 - 3t^2)\mathbf{i} - (4t^2 + 1)\mathbf{j}\) for \(t \geq 0\). Find the magnitude of the acceleration of P when \(t = 2\). [4]
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Find acceleration from velocity

A question is this type if and only if the velocity vector v(t) is given and the task requires differentiating to find the acceleration vector at a specific time or in general.

1 Moderate -0.8
1.4% of questions
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A toy remote control speed boat is launched from one edge of a small pond and moves in a straight line across the pond's surface. The boat's velocity, \(v \text{ m s}^{-1}\), is modelled in terms of time, \(t\) seconds after the boat is launched, by the expression $$v = 0.9 + 0.16t - 0.06t^2$$
  1. Find the acceleration of the boat when \(t = 2\) [3 marks]
  2. Find the displacement of the boat, from the point where it was launched, when \(t = 2\) [4 marks]
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Speed or velocity at specific time

A question is this type if and only if the task requires evaluating the velocity vector or computing the scalar speed (magnitude of velocity) at a given specific value of t.

1 Standard +0.3
1.4% of questions
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At time \(t\) seconds a particle, \(P\), has position vector \(\mathbf{r}\) metres, with respect to a fixed origin, such that $$\mathbf{r} = (t^3 - 5t^2)\mathbf{i} + (8t - t^2)\mathbf{j}$$
  1. Find the exact speed of \(P\) when \(t = 2\) [4 marks]
  2. Bella claims that the magnitude of acceleration of \(P\) will never be zero. Determine whether Bella's claim is correct. Fully justify your answer. [3 marks]
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Particle moving perpendicular to vector

A question is this type if and only if the task requires finding the time when the velocity vector is perpendicular to a given vector (or axis), by setting the dot product of velocity with that vector equal to zero.

1 Moderate -0.3
1.4% of questions
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A particle P moves so that its position vector \(\mathbf{r}\) at time \(t\) is given by $$\mathbf{r} = (5 + 20t)\mathbf{i} + (95 + 10t - 5t^2)\mathbf{j}.$$
  1. Determine the initial velocity of P. [3] At time \(t = T\), P is moving in a direction perpendicular to its initial direction of motion.
  2. Determine the value of \(T\). [3]
  3. Determine the distance of P from its initial position at time \(T\). [4]
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Particle instantaneously at rest

A question is this type if and only if the task requires finding the time(s) when both components of the velocity vector are simultaneously zero, or showing/finding the position when the particle comes to instantaneous rest.

1 Moderate -0.3
1.4% of questions
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5. At time \(t\) seconds ( \(t \geqslant 0\) ), a particle \(P\) has velocity \(\mathbf { v m ~ s } ^ { - 1 }\), where $$\mathbf { v } = \left( 3 t ^ { 2 } - 9 t + 6 \right) \mathbf { i } + \left( t ^ { 2 } + t - 6 \right) \mathbf { j }$$
  1. Find the acceleration of \(P\) when \(t = 3\) When \(t = 0 , P\) is at the fixed point \(O\).
    The particle comes to instantaneous rest at the point \(A\).
  2. Find the distance \(O A\).
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Trigonometric or exponential velocity/position

A question is this type if and only if the velocity or position vector is expressed using trigonometric (sin, cos) or exponential functions of t, requiring differentiation or integration of these non-polynomial functions.

0
0.0% of questions
1.10h Vectors in kinematics: uniform acceleration in vector form
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