Less common P3.5

Mass Weight and Terminal Velocity

This topic covers the crucial distinction between mass (the amount of 'stuff') and weight (the force of gravity on that 'stuff'). It explores how objects fall under gravity, including the effects of air resistance and the concept of a maximum speed called terminal velocity.

Part of the ESAT Physics syllabus — revision for the Engineering and Science Admissions Test (ESAT), the UAT-UK admissions test for Cambridge, Imperial, Oxford and UCL.

Key points

  • Mass is a scalar measure of an object's inertia in kilograms (kg) and is the same everywhere.
  • Weight is the vector force of gravity on an object in newtons (N), and it changes depending on the local gravitational field strength.
  • On Earth, the gravitational field strength, g, is approximated as 10 N/kg.
  • An object in free-fall (with no air resistance) accelerates downwards at a rate numerically equal to g, which is 10 m/s2 on Earth.
  • Air resistance is a force that opposes motion through the air, increasing with the object's speed and cross-sectional area.
  • Terminal velocity is reached when the upward force of air resistance exactly balances the downward force of weight, causing the net force and acceleration to become zero.

Diagram

GraphGraph with axes time / s and velocity / m s⁻¹. time / svelocity / m s⁻¹
Velocity-time graph for a falling object: the velocity rises, then levels off at the terminal velocity once air resistance grows to balance the weight.
› Why does this happen?

Why does air resistance increase with speed?

Imagine moving through the air. You have to push countless air particles out of the way. The faster you move, the more particles you have to push aside each second. Also, you have to push them aside more forcefully. These two effects combined mean that the upward force of air resistance increases as your speed increases.

Why does a falling object's acceleration decrease?

When an object first starts to fall from rest, its weight is the only significant force pulling it down. As it speeds up, an upward force of air resistance appears and gets larger. This upward force opposes the downward weight, so the resultant downward force gets smaller. According to Newton's Second Law (F=ma), a smaller resultant force causes a smaller acceleration. This means the object is still speeding up, but by a smaller amount each second.

How does this lead to terminal velocity?

Eventually, the object is falling so fast that the upward air resistance force becomes equal in size to the downward force of weight. At this point, the forces are balanced. This means the resultant force on the object is zero. According to F=ma, a zero resultant force means zero acceleration. The object can't speed up anymore. It continues to fall at a steady, maximum speed called its terminal velocity.

Formulae

w = m × g

To calculate the weight (w) of an object when you know its mass (m) and the local gravitational field strength (g).

Definitions

Mass
A measure of the amount of matter in an object, which determines its resistance to acceleration (inertia). Its unit is the kilogram (kg).
Weight
The gravitational force exerted on an object by a large body like a planet. It is a vector quantity measured in newtons (N).
Gravitational Field Strength (g)
The force of gravity exerted per unit mass at a specific location. Its units are newtons per kilogram (N/kg).
Terminal Velocity
The constant maximum velocity reached by an object falling through a fluid (like air) when the force of resistance equals the force of gravity.

Worked example

A skydiver of mass 80 kg jumps from a plane. Assume g = 10 N/kg. What is the skydiver's initial downward acceleration, and what is the magnitude of the air resistance force acting on them when they reach a terminal velocity?

  1. 1

    Step 1:

    Identify the initial situation.

    Just after jumping, the only significant force is weight, as their speed is near zero, making air resistance negligible.

  2. 2

    Step 2:

    Calculate the skydiver's weight using w = m × g.

    Weight = 80 kg × 10 N/kg = 800 N

    This is the initial resultant force.

  3. 3

    Step 3:

    Calculate the initial acceleration using Newton's second law, F = m × a.

    So, a = F / m = 800 N / 80 kg = 10 m/s2
  4. 4

    Step 4:

    Consider the situation at terminal velocity.

    The skydiver is falling at a constant speed, so their acceleration is zero.

  5. 5

    Step 5:

    If acceleration is zero, the resultant force must also be zero.

    This means the upward force of air resistance must be equal in magnitude to the downward force of weight.

  6. 6

    Step 6:

    Therefore, the air resistance force is equal to the skydiver's weight, which is 800 N.

Answer: Initial acceleration is 10 m/s2; air resistance at terminal velocity is 800 N.

Common mistakes

  • ×Using mass in kilograms instead of weight in newtons when dealing with forces. Always convert mass to weight (w = mg) first.
  • ×Stating that there are no forces on an object at terminal velocity. Instead, the forces (weight and air resistance) are balanced, resulting in zero net force.
  • ×Confusing the units: mass is in kg, weight is in N, and g is in N/kg (which is dimensionally equivalent to m/s2).

No-calculator tips

  • ✓Since g is 10 N/kg, converting between mass and weight on Earth is a simple case of multiplying or dividing by 10. For example, a 5 kg mass has a weight of 50 N.
  • ✓When an object is falling, you can quickly determine the direction of acceleration. If weight > air resistance, it accelerates downwards. If weight < air resistance (e.g. thrown down very fast), it decelerates (accelerates upwards).

Read this topic in the official UAT-UK ESAT guide →

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