5.10 Introduction

In physics, energy is a fundamental and conserved property of the universe. It is formally defined as the capacity to do work. Energy can exist in many different forms—such as mechanical, thermal (heat), chemical, electrical, and nuclear—and can be transformed from one form to another. This section focuses on kinetic energy and its relationship to work.

Derived Units→ Dimensions→

Kinetic Energy (K.E.)

Kinetic energy is the energy an object possesses due to its motion. Any object that is moving has kinetic energy.

• Definition: The energy of an object resulting from its speed.

• Formula: $K.E.=\frac{1}{2}m{v}^2$ Where:

  • $m$ is the mass of the object (kg)
  • $v$ is the speed of the object (ms${}^{-1}$)

• Key Characteristics:

  • Kinetic energy is directly proportional to the mass of the object.
  • It is proportional to the square of the object's speed. Doubling the speed quadruples the kinetic energy.
  • It is a scalar quantity and is always non-negative.

Derivation of Kinetic Energy Formula

Using Newton's second law and the equations of motion, we can derive the formula for kinetic energy.

Consider a body of mass $m$ initially at rest ($u=0$) on a frictionless surface. A constant net force $F$ acts on it over displacement $d$, giving it a final velocity $v$.

Step 1: Work done by the net force: $W=Fd$

Step 2: From Newton's second law: $F=ma$

Step 3: From the equation of motion $v^2=u^2+2ad$ with $u=0$: $v^2=2ad⟹d=\frac{v^2}{2a}$

Step 4: Substitute into the work equation: $W=ma\cdot \frac{v^2}{2a}=\frac{1}{2}m{v}^2$

Since this work is entirely converted into kinetic energy: $\boxed{K.E.=\frac{1}{2}m{v}^2}$

Kinetic Energy in Terms of Momentum

Kinetic energy can also be expressed in terms of linear momentum ($p=mv$): $K.E.=\frac{p^2}{2m}$

Derivation: Since $p=mv$, we have $v=p/m$. Substituting: $K.E.=\frac{1}{2}m{\left(\frac{p}{m}\right)}^2=\frac{p^2}{2m}$

This shows that for a fixed momentum, a lighter body has more kinetic energy.

Work-Energy Theorem

The Work-Energy Theorem states that the net work done on an object equals the change in its kinetic energy: $W_{net}=\Delta K.E.=\frac{1}{2}m{v}_f^2-\frac{1}{2}m{v}_i^2$

This theorem is a direct consequence of Newton's second law and the definition of work. It applies whether the forces are constant or variable.

In a resistive medium: When a body moves through a medium with friction or air resistance (resistive force $f$), some energy is converted to heat. For a body falling height $h$: $mgh=\frac{1}{2}m{v}^2+fh$

The term $fh$ represents the energy lost to the resistive medium (heat/sound).

Example: A 2 kg object moves at $10{\text{ ms}}^{-1}$. Its kinetic energy is: $K.E.=\frac{1}{2}(2)(10{)}^2=100\text{ J}$