Verifying given motion properties

A question is this type if and only if it asks to verify or show that a particle satisfies certain conditions (e.g., returns to origin, has specific velocity at given time) by substitution.

5 questions · Moderate -0.3

3.02f Non-uniform acceleration: using differentiation and integration
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OCR M1 2008 January Q4
8 marks Moderate -0.8
4 The displacement of a particle from a fixed point \(O\) at time \(t\) seconds is \(t ^ { 4 } - 8 t ^ { 2 } + 16\) metres, where \(t \geqslant 0\).
  1. Verify that when \(t = 2\) the particle is at rest at the point \(O\).
  2. Calculate the acceleration of the particle when \(t = 2\).
OCR M1 2013 January Q2
6 marks Moderate -0.8
2 A particle \(P\) moves in a straight line. The displacement of \(P\) from a fixed point on the line is \(\left( t ^ { 4 } - 2 t ^ { 3 } + 5 \right) \mathrm { m }\), where \(t\) is the time in seconds. Show that, when \(t = 1.5\),
  1. \(P\) is at instantaneous rest,
  2. the acceleration of \(P\) is \(9 \mathrm {~ms} ^ { - 2 }\).
AQA M2 2008 June Q6
8 marks Moderate -0.5
6 A car, of mass \(m\), is moving along a straight smooth horizontal road. At time \(t\), the car has speed \(v\). As the car moves, it experiences a resistance force of magnitude \(0.05 m v\). No other horizontal force acts on the car.
  1. Show that $$\frac { \mathrm { d } v } { \mathrm {~d} t } = - 0.05 v$$
  2. When \(t = 0\), the speed of the car is \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Show that \(v = 20 \mathrm { e } ^ { - 0.05 t }\).
  3. Find the time taken for the speed of the car to reduce to \(10 \mathrm {~ms} ^ { - 1 }\).
CAIE M1 2022 June Q6
10 marks Standard +0.3
A particle starts from a point \(O\) and moves in a straight line. The velocity \(v\) m s\(^{-1}\) of the particle at time \(t\) s after leaving \(O\) is given by $$v = k(3t^2 - 2t^3),$$ where \(k\) is a constant.
  1. Verify that the particle returns to \(O\) when \(t = 2\). [4]
  2. It is given that the acceleration of the particle is \(-13.5\) m s\(^{-2}\) for the positive value of \(t\) at which \(v = 0\). Find \(k\) and hence find the total distance travelled in the first two seconds of motion. [6]
CAIE M1 Specimen Q6
10 marks Standard +0.3
A particle \(P\) moves in a straight line, starting from a point \(O\). The velocity of \(P\), measured in m s\(^{-1}\), at time \(t\) s after leaving \(O\) is given by $$v = 0.6t - 0.03t^2.$$
  1. Verify that, when \(t = 5\), the particle is 6.25 m from \(O\). Find the acceleration of the particle at this time. [4]
  2. Find the values of \(t\) at which the particle is travelling at half of its maximum velocity. [6]