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Chapter 4: Problem 44

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 螖t is given by F = mv/螖t and (b) over distance 螖x is given by F = 2mv虏/螖x.

Step by step solution

01

Establish the basic principles

A keen understanding of Newton's second law of motion (F=ma) would be necessary. It's also important to understand that acceleration (a) can be calculated as the change in velocity (v) over the time taken (螖t), given by the formula a = v/螖t.
02

Calculate force for scenario (a)

Substituting the acceleration formula into Newton's second law of motion, the force necessary to bring the object to speed v in time 螖t can then be calculated as F = m(v/螖t), which simplifies to F = mv/螖t.
03

Establish the second principle

Another principle to consider is that velocity can also be calculated as the change in distance (螖x) over the time taken (螖t). Hence, for a constant acceleration, the final velocity can be given as v = 2a螖x.
04

Calculate force for scenario (b)

Substituting the above formula for velocity into Newton's second law of motion, the force necessary to travel the distance 螖x can then be calculated as F = m(2a螖x)/螖t. Substitute a with v/螖t (from Step 1) to get F = 2m螖x(v/螖t虏). This simplifies to F = 2mv虏/螖x.

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