FOCUSED REVIEW
Focused Review: Work, Energy, and Power
Reinforce the highest-leverage ideas and representative problem-solving tools for Work, Energy, and Power.
TIME
Approximately 15 minutes
BEST FOR
Targeted reinforcement
FINISH WITH
A readiness check
After this focused review, you’ll be able to use variable-force integrals, potential-energy derivatives, and instantaneous power relationships where they clarify the physics.
Choose how you want to review
Unit Alignment
This public Unit Review is aligned to the approved Physics Sensei mechanics architecture and is independent of textbook chapter numbering.
ARCHITECTURE: Physics Sensei Independent Mechanics
UNIT: MEC-U06 — Work, Energy, and Power
RESOURCE: Unit Review
PROFILE: Calculus-Based college physics
BEST USED
✓ Before homework on work or energy
✓ Before a quiz or exam
✓ When choosing between force-based and energy-based methods
Your Review Plan
Complete these six stages in order. Each stage builds on the previous one and prepares you for the final readiness check.
6 Stages • Approximately 15 minutes.
Warm-Up Check
Before you begin, reactivate the core ideas. Attempt each item before revealing the answer.
ACTIVITY 1
Variable-Force Work
Integrate force over position.
A one-dimensional force is F(x) = 3x² N from x = 0 to x = 2.0 m. Find the work.
Reveal Answers
8.0 J.
Why it works: W = ∫₀² 3x² dx = [x³]₀² = 8.0 J.
ACTIVITY 2
Force from Potential Energy
Differentiate the potential-energy function.
If U(x) = 4x³ J, find Fx(x).
Reveal Answers
Fx = −12x² N.
Why it works: Fx = −dU/dx.
ACTIVITY 3
Instantaneous Power
Use the force-velocity dot product.
A force F = ⟨3,4⟩ N acts while v = ⟨2,1⟩ m/s. Find instantaneous power.
Reveal Answers
10 W.
Why it works: P = F · v = 3(2) + 4(1) = 10 W.
Ready to strengthen your understanding? Now reinforce the essential concepts that control this unit.
Core Concepts
Rebuild the key energy relationships and connect each equation to its physical meaning.
KEY CONCEPT 1
Work as a Line Integral
For a variable force, work is accumulated along the displacement. In one dimension this becomes an integral of Fx(x) with respect to x.
W = ∫ F · dr; W = ∫ Fx(x) dx; Wnet = ΔK
Example: On an F-versus-x graph, signed area under the curve is work.
KEY CONCEPT 2
Potential-Energy Functions Encode Conservative Forces
A conservative force is related to the slope of the potential-energy function. Equilibrium occurs where the force is zero.
Fx = −dU/dx; F = −∇U
Example: A local minimum of U is stable equilibrium; a local maximum is unstable equilibrium.
KEY CONCEPT 3
Power Is the Time Rate of Work
Differentiating work with respect to time gives instantaneous power and the force-velocity dot product.
P = dW/dt = F · v
Example: Power can be positive, negative, or zero depending on the angle between force and velocity.
Ready to apply these ideas? Work through representative applications before the confidence check.
Guided Practice
Apply the energy model deliberately: define the system, identify the states, choose the equation, and check units and signs.
PRACTICE 1
Integrate a Variable Force
Evaluate signed work over a position interval.
F(x) = (6x − 2) N acts from x = 1.0 m to x = 4.0 m. Find the work.
Reveal Answers
39 J.
Why it works: W = ∫₁⁴(6x − 2)dx = [3x² − 2x]₁⁴ = 39 J.
PRACTICE 2
Read a Potential-Energy Function
Differentiate U(x) and interpret equilibrium.
U(x) = ax⁴ − bx² with a,b > 0. Find equilibrium positions.
Reveal Answers
x = 0 and x = ±√(b/2a).
Why it works: Set F = −dU/dx = 0 and solve 2x(2ax² − b) = 0.
PRACTICE 3
Power from Vector Data
Use the dot product at an instant.
F = ⟨5,−2,1⟩ N and v = ⟨3,4,−1⟩ m/s. Find P.
Reveal Answers
6 W.
Why it works: P = F · v = 15 − 8 − 1 = 6 W.
Ready to check your understanding? Solve the short checks without looking back at the concept cards.
Confidence Check
Use these questions to confirm that you can select and apply the correct energy model independently.
QUICK CHECK 1
Area Under F(x)
Use geometry or integration.
A force increases linearly from 0 N at x = 0 to 10 N at x = 4 m. What work is done?
Reveal Answers
20 J.
Why it works: The F-x area is a triangle: ½(4)(10) = 20 J.
QUICK CHECK 2
Potential Slope and Force
Relate dU/dx to force direction.
At a point where dU/dx > 0, what is the sign of Fx?
Reveal Answers
Negative.
Why it works: Fx = −dU/dx.
QUICK CHECK 3
Differential Work-Energy
Connect force, displacement, and kinetic energy.
What differential relation leads to the work-energy theorem?
Reveal Answers
dK = Fnet · dr.
Why it works: Integrating along the path gives ∫Fnet·dr = ΔK.
How did it go? Use the revealed explanations to identify one specific relationship to revisit if needed.
Summary
Take one final look at the most important ideas from this review.
KEY TAKEAWAY 1
Integrate Force to Get Work
Variable-force work is a path integral; in one dimension it is signed area under F(x).
KEY TAKEAWAY 2
Differentiate Potential to Get Force
Conservative force points toward decreasing potential energy.
KEY TAKEAWAY 3
Differentiate Work to Get Power
Instantaneous power is P = dW/dt = F · v.
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