Exponential displacement and model reversal
Exponential displacement and model reversal
Secondary 4
9 marks
Original parallel to supplied A Math Kinematics references
The displacement, \(s\) metres, of a particle from \(O\) after \(t\) seconds is modelled by \(s=40(1-\mathrm{e}^{-t/4})-2t\), where \(t\geq0\).
- Find the velocity and acceleration of the particle in terms of \(t\).[2]
- Find the exact time at which the particle is instantaneously at rest.[2]
- Calculate the greatest displacement of the particle from \(O\).[3]
- Comment on the suitability of this model for a particle intended to continue moving in its initial positive direction away from \(O\) for a long time.[2]
Solution:
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Answer:(i) \(v=10\mathrm e^{-t/4}-2\text{ m s}^{-1}\), \(a=-\dfrac52\mathrm e^{-t/4}\text{ m s}^{-2}\) (ii) \(t=4\ln5\text{ s}\) (iii) \(32-8\ln5\text{ m}\) (iv) Unsuitable: the particle eventually reverses and moves indefinitely in the negative direction
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