(Solved): A particle of mass \( m \) is moving along the \( x \)-axis with a velocity \( v=v(x) \) that depe ...

A particle of mass \( m \) is moving along the \( x \)-axis with a velocity \( v=v(x) \) that depends on the position \( x \) according to \[ v(x)=A \cosh (\beta x)=A\left(\frac{e^{\beta x}+e^{-\beta x}}{2}\right), \] where \( A>0 \) and \( \beta \) are constants. (a) What are the physical dimensions (units) of \( A \) and \( \beta \) ? (b) Find the force \( F(x) \) acting on the particle as a function of \( x, A \) and \( \beta \). Hint: use N2L in the form \( F=\frac{\mathrm{d} p}{\mathrm{~d} t}=m \frac{\mathrm{d} v}{\mathrm{~d} t} \) and then use the chain rule. (c) Check your expression for \( F(x) \) for dimensional consistency.