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Chapter 4: Problem 59

Although we usually write Newton's second law for one dimensional motion in the form \(F=m a,\) which holds when mass is constant, a more fundamental version is \(F=\frac{d(m v)}{d t} .\) Consider an object whose mass is changing, and use the product rule for derivatives to show that Newton's law then takes the form \(F=m a+v \frac{d m}{d t}\).

Short Answer

Expert verified
The version of Newton's second law for an object whose mass is changing is \(F = m a + v \frac{dm}{dt}\).

Step by step solution

01

Identify the Given Equation

The given equation is \(F = \frac{d(mv)}{dt}\) where \(F\) is the force, \(m\) is the mass, \(v\) is the velocity and \(t\) is the time. The mass of the object is variable.
02

Apply the Product Rule

The product rule of differentiation is given by \(\frac{d(uv)}{dt} = u\frac{dv}{dt} + v\frac{du}{dt}\) where \(u\) and \(v\) are functions of \(t\). Applying this rule to \(F = \frac{d(mv)}{dt}\), we get \(F = m\frac{dv}{dt} + v\frac{dm}{dt}\). In physics, \(\frac{dv}{dt}\) is acceleration (\(a\)), so we can replace this in the equation.
03

Finalize the expression

Substituting the value of acceleration into the expression gives us \(F = m a + v\frac{dm}{dt}\)

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