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Chapter 4: Problem 48

Find expressions for the force needed to bring an object of mass \(m\) from rest to speed \(v\) (a) in time \(\Delta t\) and (b) over distance \(\Delta x\).

Short Answer

Expert verified
The force needed to bring an object of mass m from rest to speed v (a) in time \(\Delta t\) is \(F = m \times \frac{v}{\Delta t}\), and (b) over distance \(\Delta x\) is \(F = m \times \frac{v^2}{2\Delta x}\)

Step by step solution

01

Calculate the Acceleration for part (a)

From the definition of acceleration, we have \(a = \frac{\Delta v}{\Delta t}\). As the object is initially at rest, \(\Delta v = v\). By substituting the values, we get \(a = \frac{v}{\Delta t}\)
02

Calculate the force for part (a)

We are given by Newton's second law of motion that the force \(F = ma\). Substitute the value of acceleration from step 1, to get \(F = m \times \frac{v}{\Delta t}\)
03

Calculate the acceleration for part (b)

To find acceleration, we can use the kinematic equation \(v^2 = u^2 + 2a\Delta x\). Given that the object is initially at rest \(u = 0\), we have \(a = \frac{v^2}{2\Delta x}\)
04

Calculate the force for part (b)

Using Newton's second law of motion again, we find that \(F = m \times \frac{v^2}{2\Delta x}\)

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