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Question
Answer the following question.
If the motion of an object is described by x = f(t), write formulae for instantaneous velocity and acceleration.
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Solution
- Instantaneous velocity of an object is given as,
`vec"v" = lim_(Delta"t"->0) (Delta vec"x")/(Delta"t") = ("d" vec"x")/"dt"` - Motion of the object is given as, x= f(t)
- The derivative f '(t) represents the rate of change of the position f (t) at time t, which is the instantaneous velocity of the object.
∴ `vec"v" = ("d" vec"x")/"dt" = "f"'("t")` - Acceleration is defined as the rate of change of velocity with respect to time.
- The second derivative of the position function f ''(t) represents the rate of change of velocity i.e., acceleration.
∴ `vec"a" = (Delta vec"v")/(Delta"t") = ("d"^2"x")/"dt"^2 = "f"''("t")`
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