Applications of Integration - Integration as an Inverse Process of Differentiation
Applications of Integration - Integration as an Inverse Process of Differentiation
TGM Original Questions
A particle moves in a straight line with a velocity, \(v\) m s\(^{-1}\). Its velocity can be modelled by the equation \(v = \sin t + t \cos t\). Given the initial conditions \(t = 0\), \(s = 0\).
- Differentiate \(s = t \sin t\) with respect to \(t\).
- Hence find an expression for the displacement, \(s\) at time \(t\)
- Find an expression for the acceleration, \(a\), at time \(t\).
- Determine whether the motion is periodic, and justify your answer.
- What happens to the amplitude of the motion as time progresses?
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Answer:(a) \(\sin t+t\cos t\) (b) \(s=t\sin t\) (c) \(a=2\cos t-t\sin t\) (d) Not periodic; the increasing envelope makes the continuous displacement unbounded (e) The amplitude increases linearly with \(t\)
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