1. Understand the Relationships
* Force (F): Force is the rate of change of momentum (mass times velocity). This can be expressed as: F = ma (where 'a' is acceleration, which is the rate of change of velocity).
* Energy (E): Energy is often defined as the ability to do work. Work is force times displacement. This can be expressed as: E = Fd (where 'd' is displacement).
2. Derive the Dimensions
Let's break down the dimensions using square brackets:
* [F] = [M][v]/[t] (Force is mass times acceleration, and acceleration is velocity over time)
* [E] = [F][d] = [M][v][d]/[t] (Energy is force times displacement)
3. Isolate Mass
We want to express mass ([M]) in terms of the fundamental quantities ([E], [v], [F]). We can achieve this by manipulating the equations above:
* From the force equation: [M] = [F][t]/[v]
* Substitute this expression for [M] into the energy equation: [E] = ([F][t]/[v])[v][d]/[t]
* Simplify: [E] = [F][d]
* Now, solve for [F]: [F] = [E]/[d]
* Substitute this expression for [F] back into the equation for [M]: [M] = ([E]/[d])[t]/[v]
* Final Result: [M] = [E][t]/[v][d]
Therefore, the dimensions of mass in terms of energy (E), velocity (v), force (F), and time (t) are [E][t]/[v][d].
