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Chapter 4: Problem 42

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
Part (a): The force required to bring the object from rest to speed \(v\) in time \(\Delta t\) is \(F = \frac{m \cdot v}{\Delta t}\)\n Part (b): The force required to bring the object from rest to speed \(v\) over distance \(\Delta x\) is \(F = m \cdot \sqrt { \frac{v^2}{2 \Delta x}}\).

Step by step solution

01

Calculate the change in momentum

The first step in solving this exercise is to calculate the change in momentum, which can be calculated using the formula \(\Delta p = m \cdot v\) since the object is initially at rest.
02

Calculate force required using time

The force required to bring the object from rest to speed \(v\) in time \(\Delta t\) can be calculated using the formula \(F = \Delta p / \Delta t\). Substitute the value of \(\Delta p\) from Step 1 into this formula to find \(F\).
03

Calculate acceleration

Acceleration \(a\) is the change in velocity over time. So, find it by rearranging the formula \(F = m \cdot a\) after substituting the value for \(F\) from Step 2.
04

Calculate force required using distance

Using \( v^2 = u^2 + 2a \Delta x \), solve for \(a\). Next, substitute this value for \(a\) into the formula \(F = m \cdot a\) to calculate the force required to bring the object from rest to speed \(v\) over distance \(\Delta x\).

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