Density tells us how much mass is packed into a given volume. It describes how compact a substance is.

$\rho = \frac{m}{V}$

Where:

• $\rho$ = density (kg/m³ or g/cm³)
• $m$ = mass (kg or g)
• $V$ = volume (m³ or cm³)

The SI unit of density is kg/m³. A useful comparison: water has a density of 1000 kg/m³ or 1 g/cm³.

Example: A block of copper has a mass of 890 g and a volume of 100 cm³. Find its density.

$\rho = \frac{m}{V} = \frac{890 \, \text{g}}{100 \, \text{cm}^3} = 8.9 \, \text{g/cm}^3$

This tells us copper is a heavy material—nearly 9 times denser than water.

Pressure is the force applied per unit area. When a force acts on a smaller area, the pressure is greater.

$P = \frac{F}{A}$

Where:

• $P$ = pressure (Pascals, Pa or N/m²)
• $F$ = force (Newtons, N)
• $A$ = area (m²)

Example: A student weighing 500 N stands on the floor. If the total area of both shoes in contact with the ground is 0.04 m², what pressure is exerted?

$P = \frac{500 \, \text{N}}{0.04 \, \text{m}^2} = 12,500 \, \text{Pa}$

If the student stands on one foot, the area halves and pressure doubles. This is why sharp objects (small area) can easily pierce surfaces despite small forces.

Velocity describes how fast something moves in a particular direction.

$v = \frac{d}{t}$

Where:

• $v$ = velocity (m/s)
• $d$ = distance (m)
• $t$ = time (s)

Example: A motorcycle travels 150 m along a road in 10 seconds. What is its velocity?

$v = \frac{150 \, \text{m}}{10 \, \text{s}} = 15 \, \text{m/s}$

Acceleration is the rate at which velocity changes. If an object speeds up, slows down, or changes direction, it is accelerating.

$a = \frac{v - u}{t}$

Where:

• $a$ = acceleration (m/s²)
• $v$ = final velocity (m/s)
• $u$ = initial velocity (m/s)
• $t$ = time (s)

Example: A bicycle starts from rest and reaches a speed of 8 m/s in 4 seconds. Find its acceleration.

$a = \frac{8 \, \text{m/s} - 0 \, \text{m/s}}{4 \, \text{s}} = \frac{8}{4} = 2 \, \text{m/s}^2$

When a force acts on a mass, it produces acceleration. The relationship is:

$F = m \times a$

Where:

• $F$ = force (N)
• $m$ = mass (kg)
• $a$ = acceleration (m/s²)

Example: A loaded tractor of mass 2000 kg accelerates at 0.5 m/s². What force is the engine producing?

$F = 2000 \, \text{kg} \times 0.5 \, \text{m/s}^2 = 1000 \, \text{N}$

When a force causes motion, work is done. Work is the product of force and the distance moved in the direction of the force.

$W = F \times d$

Where:

• $W$ = work (Joules, J)
• $F$ = force (N)
• $d$ = distance (m)

Example: A porter pushes a wheelbarrow with a force of 150 N through a distance of 6 m. Calculate the work done.

$W = 150 \, \text{N} \times 6 \, \text{m} = 900 \, \text{J}$

Power measures how quickly work is done—the rate of doing work.

$P = \frac{W}{t}$

Where:

• $P$ = power (Watts, W)
• $W$ = work (J)
• $t$ = time (s)

Example: A small electric pump does 3000 J of work in 15 seconds. What is its power?

$P = \frac{3000 \, \text{J}}{15 \, \text{s}} = 200 \, \text{W}$

In Tanzania, understanding these relationships helps in everyday situations. For example, when buying charcoal or rice in the market, vendors use the concept of density—selling by volume (a bucket) rather than mass, since people know that 1 litre of charcoal has a consistent weight. Similarly, knowing that force divided by area gives pressure helps students understand why a mata (machete) has a sharp thin blade: the same force applied over a very small area produces very high pressure, making it easy to cut firewood or crops.