5.1.5Work, energy, power and deformation
Derivation of P = Fv
5 min read 6 parts Lesson 8 of 17 1 quiz
Lesson summary
5.1.5 Derivation of P = Fv — AS PHYSICS 9702 notes: Derive P = Fv from the definitions of work and power and state when it applies.Result — P = Fv gives the power delivered to a moving object.P is power in watts (W), F is force in newtons (N), v is velocity in metres per second (…
- Conditions
- What the symbols mean
- Force needed
- Drag force
- Inclined road
- Component of force
Section 1 of 6The equation
Result
- P = Fv gives the power delivered to a moving object.
- P is power in watts (W), F is force in newtons (N), v is velocity in metres per second (m s−1).
Conditions
- Conditions a constant force moves a body at constant velocity.
- The force must act in the same direction as the velocity.
- Power is delivered to the object by the force in order to produce motion.
Section 2 of 6Derivation
What the symbols mean:
Power is the rate of doing work, P = W/t.
What the symbols mean: P = power (W = J s−1) · W or E = work done / energy transferred (J) · t = time (s) · F = force (N) · v = velocity (m s−1)
What the symbols mean:
Work is done when a force moves an object over a distance, W = Fs.
What the symbols mean: W = work done = energy transferred (J) · F = force (N) · d, s or x = distance moved in the direction of the force (m)
At constant velocity the distance moved is s = vt.
Substituting gives W = Fvt.
Substituting into the power equation gives P = Fvt/t.
Cancelling t gives P = Fv.
Section 3 of 6Using P = Fv
Using P = Fv
Force needed
- Force needed the force in the equation is the force applied by the engine or motor.
- At constant speed it equals the resistive force it must overcome.
Drag force
- Drag force a drag force acts opposite to the velocity, but the force used to calculate power is equal to (or above) the drag force.
Inclined road
- Inclined road at constant speed up a slope, the force to overcome weight is F = mg sin θ.
- The useful power is then P = (mg sin θ)v, plus any resistive force × v.
Component of force
- Component of force if the force is not parallel to the velocity, only the component along the velocity is used.
Section 4 of 6More from the specification
More from the specification
Derivation of Mechanical Power Equation P = Fv
- Power is defined as the rate of energy transferred or work done over time;
- (P = W / t = F·s / t)
- since velocity is v = s / t, the instantaneous power supplied by a driving force F moving an object at velocity v is P = F·v.
Energy Exchanges in Dynamic Systems
- In mechanical systems like hydroelectric turbines or roller coasters, gravitational potential energy stored in elevated masses is transferred into a kinetic energy store as it descends (mgh → ½mv²).
- (such as reservoir water)
Efficiency and Dissipated Energy
- In real mechanical systems, frictional resistance causes a portion of input power to be lost (Plost) as wasted thermal energy, yielding an operational efficiency η = × 100%.
- (useful power output / total power input)
Efficiency
- Useful output divided by total input.
Thermal
- A convection current in the air.
Energy store
- Where energy is held, such as kinetic or chemical.
Gravitational potential
- Work done per unit mass in bringing a test mass from infinity to the point.
Kinetic energy
- K = ½mv2, energy of motion.
Output power
- Useful rate of energy delivered by a device.
Thermal store
- Energy held in the moving particles of a material.
Mains supply
- Domestic a.c. supply, rated 230 V rms at 50 Hz in the UK.
Section 5 of 6Key terms
| Term | Meaning |
|---|---|
| Power | Rate of doing work, P = W/t. |
| P = Fv | Power equals force multiplied by velocity for a constant force along constant velocity. |
| Driving force | Force applied by an engine or motor to produce motion. |
| Drag force | Resistive force opposing motion through a fluid. |
| Component of weight | Part of weight acting along an incline, mg sin θ. |
Section 6 of 6Exam technique
1
Using P = Fv when the force is not along the velocity — only the component along the motion counts.
2
Using the resultant force instead of the driving force.
3
Forgetting to resolve weight down the slope (mg sin θ) for a vehicle on an incline.
4
Using P = Fv with non-constant velocity as an average — it gives the instantaneous power.