Why does mass cancel in free-fall energy problems?

Because both kinetic energy (½mv²) and potential energy (mgh) are proportional to mass. When you equate them, m appears on both sides and cancels. This is why Galileo was right: in the absence of air resistance, all objects fall at the same rate regardless of mass.

What happens to energy when friction is present?

Friction converts mechanical energy into thermal energy (heat). The total energy of the universe is still conserved, but the useful mechanical energy decreases. This is why perpetual motion machines are impossible — friction always dissipates some energy.

Is potential energy always gravitational?

No. Potential energy is stored energy associated with position in any force field. Elastic potential energy (PE = ½kx²) is stored in a compressed spring. Electric potential energy is stored between charged particles. Chemical potential energy is stored in molecular bonds.

What is the unit of energy and why is it called a joule?

The SI unit of energy is the joule (J), named after physicist James Prescott Joule. 1 J = 1 N·m = 1 kg·m²/s². In everyday terms: lifting a 100 g apple by 1 metre requires about 1 joule of energy.

Kinetic and Potential Energy Visualiser

Energy is the central concept of physics. Everything that happens — a ball rolling down a hill, a rocket launching, an atom emitting light — involves energy changing from one form to another. The two most fundamental mechanical forms are kinetic energy (energy of motion) and potential energy (energy of position).

The law of conservation of energy says the total mechanical energy stays constant in the absence of non-conservative forces like friction. This principle underlies projectile motion and every dynamical system governed by Newton's Laws. Watch it in action as a mass descends a ramp.

Total energy 490.5 J, PE at the top 490.5 J. At the bottom: KE 490.5 J, speed 14.0 m/s.

Potential energy is the lower yellow area, kinetic energy the upper one, so the top edge is the total. With friction, 20% of the starting energy is lost across the full ramp.

Mass5 kg

Initial height10 m

Initial speed0 m/s

Include friction (20% loss)

The Formulas

Kinetic Energy

$KE = \frac{1}{2}mv^2$

Kinetic energy depends on mass and the square of velocity. Doubling the speed quadruples the kinetic energy.

Gravitational Potential Energy

$PE = mgh$

Potential energy depends on mass, gravitational acceleration g (9.81 m/s²), and height above the reference level.

Conservation of Mechanical Energy

In the absence of friction:

$KE_1 + PE_1 = KE_2 + PE_2$

$\frac{1}{2}mv_1^2 + mgh_1 = \frac{1}{2}mv_2^2 + mgh_2$

Symbol	Quantity	Unit
m	Mass	kg
v	Speed	m/s
g	Gravitational acceleration	m/s² (9.81 on Earth)
h	Height	m
KE	Kinetic energy	J (joule)
PE	Potential energy	J (joule)

Worked Examples

Worked example

Example 1 — Ball dropped from rest

A 2 kg ball is dropped from a height of 20 m. What is its speed just before hitting the ground? (Ignore air resistance.)

Using conservation of energy:

At the top: all energy is PE = mgh = 2 × 9.81 × 20 = 392.4 J, KE = 0

At the bottom: all energy is KE, PE = 0

$\frac{1}{2}mv^2 = mgh \quad \Rightarrow \quad v = \sqrt{2gh} = \sqrt{2 \times 9.81 \times 20} \approx 19.8 \text{ m/s}$

Note: the mass cancels — all objects (regardless of mass) reach the same speed when dropped from the same height.

Worked example

Example 2 — Roller coaster minimum speed

A roller coaster cart starts from rest at a height of 40 m. What is the minimum height a loop of radius 8 m can be at for the cart to complete the loop?

At the top of the loop, the cart needs a minimum centripetal acceleration of g:

$v_{min}^2 = gR = 9.81 \times 8 = 78.5 \text{ m}^2\text{/s}^2$

Using energy conservation from start height $H$ to loop top height $h_{loop} + 2R$ :

$mgH = mg(h_{loop} + 2R) + \frac{1}{2}mv_{min}^2$

If the loop sits at ground level ( $h_{loop} = 0$ ):

$h_{loop} = H - 2R - \frac{v_{min}^2}{2g} = 40 - 16 - 4 = 20 \text{ m}$

The loop top is at 16 m. Since 40 m > 16 m, the cart completes the loop with speed to spare.

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