If the velocity of an object is given by # v(t) = 3t^2 - 22t + 24 # and #s(0)=0# then how do you find the displacement at time #t#?

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Answer 1 · 2017-11-08T11:23:04

# s = t^3 - 11t^2 + 24t #

We have:

# v(t) = 3t^2 - 22t + 24 # and #s(0)=0# ..... [A]

We know that:

# v = (ds)/dt #

So we can write [A] as a Differential Equation:

# (ds)/dt = 3t^2 - 22t + 24 #

This is separable. "as is", so we can "separate the variables" to get:

# int \ ds = int \ 3t^2 - 22t + 24 \ dt #

Which we can directly integrate, to get:

# s = t^3 - 11t^2 + 24t + C #

Using the initial condition #s(0)=0# we have:

# 0 = 0 - 0 + 0 + C => C=0#

Giving uis a position function for the particle at time #t#:

# s = t^3 - 11t^2 + 24t #