A2 UNIT 1

A. Space, time and motion

A.3 Work, energy and power

DP IB Physics: SL A. Space, time and motion A.3 Work, energy and power Understandings Students should understand:
a)	The principle of the conservation of energy
b)	That work done by a force is equivalent to a transfer of energy
c)	That energy transfers can be represented on a Sankey diagram
d)	That work W done on a body by a constant force depends on the component of the force along the line of displacement as given by [math]W = Fs cos θ[/math]
e)	That work done by the resultant force on a system is equal to the change in the energy of the system
f)	That mechanical energy is the sum of kinetic energy, gravitational potential energy and elastic potential energy
g)	That in the absence of frictional, resistive forces, the total mechanical energy of a system is conserved
h)	That if mechanical energy is conserved, work is the amount of energy transformed between different forms of mechanical energy in a system, such as: – The kinetic energy of translational motion as given by [math]E_k = \frac{1}{2}mv^2 = \frac{p^2}{2m}[/math] the gravitational potential energy, when close to the surface of the Earth as given by [math]ΔE_p = mgΔh[/math] – The elastic potential energy as given by [math]E_H = \frac{1}{2}k(\Delta x)^2[/math]
i)	That power developed P is the rate of work done, or the rate of energy transfer, as given by [math]P = \frac{\Delta W}{\Delta t} = Fv[/math]
j)	Efficiency η in terms of energy transfer or power as given by [math]\eta = \frac{E_{\text{output}}}{E_{\text{input}}} = \frac{P_{\text{output}}}{P_{\text{input}}}[/math]
k)	Energy density of the fuel sources.

(a) The Principle of Conservation of Energy
The principle of conservation of energy states that energy cannot be created or destroyed; it can only be transferred or converted from one form to another. The total energy in a closed system remains constant.
Total Energy Before = Total Energy After
Figure 1 Principle of conservation of energy
Energy can transform between forms:
– Kinetic Energy (KE) → Potential Energy (PE) (e.g., a ball thrown upwards)
– Chemical Energy → Thermal Energy (e.g., burning fuel)
– Electrical Energy → Light + Heat (e.g., a light bulb)
Some energy is often lost as heat due to friction, air resistance, or inefficiencies.
⇒ Example:
A roller coaster at the highest point has maximum gravitational potential energy (GPE).
As it moves down, GPE is converted into kinetic energy (KE).
At the lowest point, KE is maximum.
Some energy is lost as heat and sound due to friction.
⇒ Types of energy

Form of Energy	Define
Kinetic	The energy of a moving object
Gravitational Potential	The energy something gains when you lift it up, and which it loses when it falls.
Elastic	The energy of a stretched spring or elastic band. (Sometimes called strain energy)
Chemical	The energy contained in a chemical substance.
Nuclear	The energy contained within the nucleus an atom
Internal	The energy something has due to its temperature (or state), (sometimes referred to as thermal or heat energy)

(b) Work Done by a Force is Equivalent to a Transfer of Energy
The work done by a force is the amount of energy transferred when a force moves an object. It is calculated using:
[math]W = Fdcosθ[/math]
Where:
– W = Work done (Joules, J)
– F = Applied force (Newtons, N)
– d = Distance moved in the direction of force (meters, m)
– θ = Angle between force and displacement
Work done = Energy transferred (in the same units,Joules).
Figure 2 Work done by a force
Positive work: When force and displacement are in the same direction (e.g., lifting an object).
Negative work: When force opposes displacement (e.g., friction slowing down a car).
⇒ Example:
A person pushes a box with a force of 50 N over a distance of 4 m.
[math]\text{Work done} = F \cdot d \\
\text{Work done} = 50 \times 4 \\
\text{Work done} = 200\ \text{J}[/math]
This 200J of energy is transferred to the box as kinetic energy.
(c) Energy Transfers Represented on a Sankey Diagram
A Sankey diagram is a visual representation of how energy is distributed, used, and wasted in a system.
– The width of each arrow represents the amount of energy.
– Useful energy is shown going forward, while wasted energy branches off.
Figure 3 Energy transfers represented on a Sankey diagram
⇒ Example: A Light Bulb
– 100 J of electrical energy input
– 10 J converted to useful light energy
– 90 J lost as heat
A Sankey diagram for this would look like:
Thick arrow (100 J input) → Splitting into 10 J (useful light) + 90 J (wasted heat).
⇒ Applications of Sankey Diagrams
Helps identify inefficiencies in machines and energy systems.
Useful in improving energy efficiency in homes, industries, and transportation.
(d) Work Done by a Constant Force
Work W is done when a force moves an object through a distance. The amount of work depends on:
– The magnitude of the force applied
– The distance moved in the direction of the force
– The angle between the force and the displacement
Formula for Work Done
[math]W = Fscosθ[/math]
Where:
– W = Work done (Joules, J)
– F = Applied force (Newtons, N)
– s = Displacement (meters, m)
– θ = Angle between force and displacement
Figure 4 Work done by a constant force
When [math]θ = 0^0[/math] (Force is parallel to displacement):
W = Fs (Maximum work is done)
When [math]θ = 90^0[/math] (Force is perpendicular to displacement):
W = 0 (No work is done,e.g.,carrying a bag without lifting it)
When [math]θ = 180^0[/math] (Force opposes displacement):
– W is negative, meaning energy is removed from the system (e.g., friction slowing an object).
⇒ Example
A box is pushed with a force of 50 N at an angle of 30° to the horizontal. It moves 5 m forward
[math]W = F s \cos \theta \\
W = (50)(5) \cos 30^\circ \\
W = (50)(5)(0.866) \\
W = 216.5\ \text{J}[/math]
So, 216.5 J of energy is transferred to the box.
(e) Work Done by the Resultant Force and Energy Change
⇒ Work-Energy Theorem
Work Done by the Resultant Force = Change in Energy
This means that if a net force acts on a system and does work, the system’s energy increases or decreases.
Figure 5 Work- Energy Theorem
– If work is positive → Energy increases (e.g., pushing a car).
– If work is negative → Energy decreases (e.g., applying brakes).
Formula for Work and Energy Change
[math]W = \Delta KE \\
W = \frac{1}{2}mv^2 – \frac{1}{2}mu^2[/math]
Where:
– m = mass of object
– v = final velocity
– u = initial velocity
⇒ Example
A 2 kg object speeds up from 3 m/s to 6 m/s due to a net force:
[math]W = \frac{1}{2}mv^2 – \frac{1}{2}mu^2 \\
W = \frac{1}{2}(2)(6)^2 – \frac{1}{2}(2)(3)^2 \\
W = 36 – 9 \\
W = 27\ \text{J}[/math]
So, 27 J of energy is added to the object.
(f) Mechanical Energy
⇒ Definition
Mechanical energy is the sum of all kinetic, gravitational potential, and elastic potential energy in a system. It is given by:
[math]E_{\text{mechanical}} = KE + PE_g + PE_e[/math]
Where:
– Kinetic Energy (KE): Energy due to motion
[math]KE = \frac{1}{2}mv^2[/math]
– Gravitational Potential Energy ([math]PE_g[/math]): Energy due to height
[math]PE_g = mgh[/math]
– Elastic Potential Energy ( [math]PE_e[/math]): Energy stored in stretched or compressed elastic materials
[math]PE_e = \frac{1}{2}mx^2[/math]
(where k is the spring constant and x is the displacement from equilibrium)
Figure 6 Mechanical Energy
⇒ Conservation of Mechanical Energy
If there is no external force (like friction), total mechanical energy remains constant:
[math]KE_{\text{initial}} + PE_{\text{initial}} = KE_{\text{final}} + PE_{\text{final}}[/math]
⇒ Example
A 2 kg ball is dropped from a height of 10 m. Find its speed just before hitting the ground.
– Initial energy:
[math]PE = mgh \\
PE = (2)(9.8)(10) \\
PE = 196\ \text{J}[/math]
– At ground level,
[math]KE = \frac{1}{2}mv^2 \\
196 = \frac{1}{2}(2)v^2 \\
v^2 = 196 \\
v = 14\ \text{m/s}[/math]
So, the ball hits the ground at 14 m/s.
(g) Conservation of Mechanical Energy and Energy Transformations
⇒ Conservation of Mechanical Energy
In a frictionless system, the total mechanical energy (TME) remains constant:
[math]E_{\text{mechanical}} = KE + PE[/math]
Where:
KE = Kinetic Energy (Energy due to motion)
PE = Potential Energy (Stored energy due to position or deformation)
This principle states that energy cannot be created or destroyed, only transformed between different forms of mechanical energy.
Figure 7 Conservation of Mechanical energy
(h) Energy Transformations in a System
Work done in a system without resistive forces converts energy from one form to another:
– Kinetic Energy (KE) – Translational Motion
The energy of a moving object is given by:
[math]E_k = \frac{1}{2}mv^2[/math]
Or in terms of momentum:
[math]E_k = \frac{p^2}{2m}[/math]
Where:
m = Mass of the object (kg)
v = Velocity of the object (m/s)
p = Momentum ([math]p = mv[/math])
Figure 8 Energy transformation in system
⇒ Example:
A 2 kg object moves at 5 m/s. Find its kinetic energy.
[math]E_k = \frac{1}{2}mv^2 \\
E_k = \frac{1}{2}(2)(5)^2 \\
E_k = 25\ \text{J}[/math]
Energy Transformation:
If the object moves uphill, KE converts to gravitational potential energy.
⇒ Gravitational Potential Energy (GPE) – Energy Due to Height
Close to the Earth’s surface, gravitational potential energy is given by:
[math]ΔE_p = mgΔh[/math]
Where:
m = Mass of the object (kg)
g = Acceleration due to gravity ([math]9.81m/s^2[/math])
[math]Δh[/math] = Change in height (m)
⇒ Example:
A 3 kg object is lifted 10 m above the ground. Find its GPE.
[math]\Delta E_p = mg \Delta h \\
\Delta E_p = (3)(9.81)(10) \\
\Delta E_p = 294.3\ \text{J}[/math]
Energy Transformation:
If the object is dropped, GPE converts into KE as it falls.
– Elastic Potential Energy (EPE) – Stored Energy in Springs or Elastic Materials
When a spring or elastic object is stretched or compressed, it stores energy:
[math]E_H = \frac{1}{2}k(\Delta x)^2[/math]
Where:
k = Spring constant (N/m)
[math]\Delta x[/math] = Extension or compression of the spring (m)
⇒ Example:
A spring with k=200 N/m is stretched by 0.05 m. Find its stored elastic potential energy.
[math]E_H = \frac{1}{2}k(\Delta x)^2 \\
E_H = \frac{1}{2}(200)(0.05)^2 \\
E_H = 0.25\ \text{J}[/math]
Energy Transformation:
When released, EPE converts into KE, causing the spring to move.
⇒ Conservation of Energy in a Free-Fall System
Let’s consider a ball dropped from a height h with zero initial velocity.
⇒ At the top (Before falling):
KE = 0
PE = mgh
⇒ Midway (During fall):
Some PE is converted into KE
[math]E_{\text{mechanical}} = KE + PE[/math] remains constant
⇒ Just before hitting the ground:
KE is maximum
PE=0
KE = mgh
This shows that gravitational potential energy is fully converted into kinetic energy.
i) Power: The Rate of Energy Transfer
Power (P) is defined as the rate at which work is done or energy is transferred. It is given by:
[math]P = \frac{\Delta W}{\Delta t}[/math]
Where:
P = Power (Watts, W)
ΔW = Work done or energy transferred (Joules, J)
Δt = Time taken (seconds, s)
⇒ Alternative Formula (For a Moving Object):
Since work done is given by:
[math]W = Fs[/math]
Dividing by time (t):
[math]P = \frac{Fs}{t}[/math]
Since velocity [math]v = \frac{s}{t}[/math], we can write:
[math]P = Fv[/math]
Where:
F = Force (Newtons, N)
v = Velocity (m/s)
If a constant force is applied to move an object at velocity v, power is directly proportional to force and velocity.
In engines, higher power means faster acceleration.
Figure 9 The rate of energy transfer
⇒ Example 1: Power Calculation
A car engine applies a force of 4000 N to move a car at 30 m/s. Find the power.
[math]P = Fv \\
P = (4000)(30) \\
P = 120,000\ \text{W} \\
P = 120\ \text{kW}[/math]
⇒ Example 2: Work-Based Power Calculation
A crane lifts a 500 kg object 20 m in 10 seconds. Find the power.
[math]W = mgh \\
W = (500)(9.81)(20) \\
W = 98,100\ \text{J} \\
p = \frac{W}{t} \\
p = \frac{98,100}{10} \\
p = 9,810\ \text{W} \\
p = 9.81\ \text{kW}[/math]
j) Efficiency: Energy or Power Conversion
Efficiency (η) is a measure of how much input energy is converted into useful output energy.
[math]\eta = \frac{E_{\text{output}}}{E_{\text{input}}} \times 100\%[/math]
Or in terms of power:
[math]\eta = \frac{P_{\text{output}}}{P_{\text{input}}} \times 100\%[/math]
Where:
[math]E_{\text{output}}[/math] = Useful energy output (J)
[math]E_{\text{input}}[/math] = Total energy input (J)
[math]P_{\text{output}}[/math] = Useful power output (W)
[math]P_{\text{input}}[/math] = Total power input (W)
Figure 10 Efficiency: Energy or power conversion
⇒ Example 1: Efficiency Calculation
An electric motor receives 2000 J of energy but produces 1500 J of useful energy.
[math]\eta = \frac{E_{\text{output}}}{E_{\text{input}}} \times 100\% \\
\eta = \frac{1500}{2000} \times 100\% \\
\eta = 75\%[/math]
⇒ Example 2: Efficiency of a Power Plant
A power station generates 600 MW of electricity but uses 1800 MW of fuel energy.
[math]\eta = \frac{P_{\text{output}}}{P_{\text{input}}} \times 100\% \\
\eta = \frac{600}{1800} \times 100\% \\
\eta = 33.3\%[/math]
Efficiency is always ≤ 100% because some energy is lost as heat, sound, or friction.
Higher efficiency means less energy waste and better performance.
k) Energy Density: The Energy Stored in Fuels
Energy density is the amount of energy stored per unit mass or volume of a fuel. It is given by:
[math]\text{Energy Density} = \frac{\text{Energy Released}}{\text{Mass of Fuel}}[/math]
Where:
Units: J/kg (for mass-based) or J/m³ (for volume-based).
⇒ Energy Density of Common Fuels:

Fuel Type	Energy Density (J/kg)
Gasoline	[math]4.6 × 10^7[/math]
Diesel	[math]4.8 × 10^7[/math]
Coal	[math]2.4 × 10^7[/math]
Natural Gas	[math]5.5 × 10^7[/math]
Hydrogen	[math]1.2 × 10^7[/math]

⇒ Example: Calculating Energy from Fuel
A car burns 5 kg of gasoline. How much energy is released?
[math]\text{Energy Density} = \frac{\text{Energy Released}}{\text{Mass of Fuel}} \\
\text{Energy Released} = \text{Energy Density} \times \text{Mass of Fuel} \\
E = (4.6 \times 10^7) \times 5 \\
E = 2.3 \times 10^8\ \text{J}[/math]
Fuels with higher energy density provide more energy per unit mass.
Hydrogen has the highest energy density, but it is difficult to store.
Renewable sources like batteries have lower energy densities compared to fossil fuels.

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A2 UNIT 1

A. Space, time and motion

DP IB Physics: SL

A. Space, time and motion

A.3 Work, energy and power

(a) The Principle of Conservation of Energy

(b) Work Done by a Force is Equivalent to a Transfer of Energy

(c) Energy Transfers Represented on a Sankey Diagram

(d) Work Done by a Constant Force

(e) Work Done by the Resultant Force and Energy Change

(f) Mechanical Energy

(g) Conservation of Mechanical Energy and Energy Transformations

(h) Energy Transformations in a System

i) Power: The Rate of Energy Transfer

j) Efficiency: Energy or Power Conversion

k) Energy Density: The Energy Stored in Fuels

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