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Calculate the distance a particle is moved by a given force function

by RoRi

Considering the force function f(x) = ax^2 + bx propelling a particle along the x-axis. Find a and b such that 900 ergs of work are done to move the particle 10cm from (0,0) if f(5) = 65.

We are given f(5) = 65 so,

    \[ f(5) = 25a + 5b = 65 \quad \implies \quad b = 13 -5a. \]

Then, since 900 ergs of work are required to move the particle 10cm we have

    \begin{align*} \int_0^{10} (ax^2+bx) \, dx = 900 && \implies && \left. \frac{ax^3}{3} \right|_0^{10} + \left. \frac{bx^2}{2} \right|_0^{10} &= 900 \\ && \implies && \left(\frac{1000}{3} \right)a + 50 b &= 900 \\ && \implies && \left( \frac{1000}{3} \right)a + 650 - 250a &= 900 \\ && \implies && a = 3, \quad b = -2. \end{align*}

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