06Unit 2 Review
Consolidate your understanding of circular motion, gravity, momentum, energy, and special relativity with mixed conceptual questions, calculation problems, and a student self-check.
Consolidating Unit 2 prepares you for thermal physics and waves in Unit 3, where energy concepts reappear in new contexts.
How do the laws of motion, gravity, energy, and relativity work together to describe the behavior of objects — from a spinning satellite to a particle traveling near the speed of light?
Unit 2 Summary
Ch 1 · Circular Motion
Objects moving in a circle require a net inward (centripetal) force. Centripetal acceleration equals v²/r, and the centripetal force equals mv²/r. Period, frequency, and angular velocity describe how fast an object completes each revolution.
Ch 2 · Gravity and Orbital Mechanics
Newton's law of universal gravitation states that every mass attracts every other mass with a force proportional to the product of their masses and inversely proportional to the square of the distance between them (Fg = Gm₁m₂/r²). Orbital speed and period follow directly from setting gravitational force equal to centripetal force.
Ch 3 · Momentum and Collisions
Momentum (p = mv) is conserved in all closed systems. Impulse (J = FΔt) equals the change in momentum. Elastic collisions conserve both momentum and kinetic energy; inelastic collisions conserve only momentum.
Ch 4 · Work, Energy, and Simple Machines
Work equals force times displacement times the cosine of the angle between them (W = Fd cosθ). The work-energy theorem states that net work equals the change in kinetic energy. Mechanical energy (KE + PE) is conserved when only conservative forces act. Simple machines trade force for distance while conserving work.
Ch 5 · Special Relativity
At speeds approaching c, time dilates (t′ = γt), lengths contract (L′ = L/γ), and mass-energy equivalence holds (E = mc²). The Lorentz factor γ = 1/√(1 − v²/c²) quantifies relativistic effects.
Key Equations
Fc = mv²/r
Centripetal force
ac = v²/r
Centripetal acceleration
Fg = Gm₁m₂/r²
Newton's gravitation
p = mv
Momentum
J = FΔt = Δp
Impulse
W = Fd cosθ
Work
KE = ½mv²
Kinetic energy
PE = mgh
Gravitational PE
ME = KE + PE
Mechanical energy
t' = γt
Time dilation
L' = L/γ
Length contraction
E = mc²
Mass-energy
Worked Examples
A 1,200 kg car rounds a flat circular curve of radius 80 m at 20 m/s. What centripetal force does the road exert on the car?
Identify: m = 1,200 kg, v = 20 m/s, r = 80 m
Formula: Fc = mv²/r
Fc = (1,200)(20²) / 80
Fc = (1,200)(400) / 80
Fc = 480,000 / 80
Calculate the gravitational force between Earth (m₁ = 5.97 × 10²⁴ kg) and a 70 kg person standing on its surface (r = 6.37 × 10⁶ m). G = 6.674 × 10⁻¹¹ N·m²/kg².
Formula: Fg = Gm₁m₂/r²
Fg = (6.674 × 10⁻¹¹)(5.97 × 10²⁴)(70) / (6.37 × 10⁶)²
Numerator: 6.674 × 10⁻¹¹ × 4.179 × 10²⁶ ≈ 2.789 × 10¹⁶
Denominator: (6.37 × 10⁶)² = 4.058 × 10¹³
Fg ≈ 2.789 × 10¹⁶ / 4.058 × 10¹³
A 3 kg ball moving at 6 m/s east collides with a stationary 5 kg ball. After the collision the 3 kg ball moves at 1 m/s east. Find the velocity of the 5 kg ball after the collision.
Conservation of momentum: p_before = p_after
p_before = (3)(6) + (5)(0) = 18 kg·m/s
p_after = (3)(1) + (5)v₂
18 = 3 + 5v₂
5v₂ = 15
A 2 kg block starts from rest and slides down a frictionless ramp of height 5 m. What is its speed at the bottom? (g = 9.8 m/s²)
Conservation of mechanical energy: PE_top = KE_bottom
mgh = ½mv²
gh = ½v² (mass cancels)
v² = 2gh = 2(9.8)(5) = 98
v = √98
A muon created in the upper atmosphere has a proper lifetime of 2.2 μs. It travels at v = 0.98c relative to Earth. How long does the muon's lifetime appear to an observer on Earth? (γ = 1/√(1 − 0.98²) ≈ 5.03)
Time dilation: t′ = γt₀
t₀ = 2.2 × 10⁻⁶ s (proper lifetime in muon frame)
γ ≈ 5.03
t' = 5.03 × 2.2 × 10⁻⁶
Guided Practice
A 0.5 kg ball on a 1.2 m string is swung in a horizontal circle at 4 m/s. What is the tension in the string?
Hint: Tension provides the centripetal force. Use Fc = mv²/r with r = 1.2 m.
Two asteroids have masses 4.0 × 10¹² kg and 6.0 × 10¹² kg and are separated by 500 m. Find the gravitational force between them. (G = 6.674 × 10⁻¹¹ N·m²/kg²)
Hint: Apply Fg = Gm₁m₂/r². Square the distance first: r² = (500)² = 250,000 m².
A 0.2 kg hockey puck slides at 8 m/s and is brought to rest by a 4 N friction force. How long does the friction act? Use the impulse-momentum theorem.
Hint: J = FΔt = Δp. The change in momentum is 0 − (0.2)(8) = −1.6 kg·m/s. Solve for Δt.
A 10 kg crate is pushed 6 m along a floor by a 50 N force applied at 30° below the horizontal. How much work is done by the applied force?
Hint: W = Fd cosθ. The angle between force and displacement is 30°. cos 30° ≈ 0.866.
A spaceship travels at v = 0.60c. Its proper length is 200 m. What length does a stationary observer measure? (γ = 1.25)
Hint: Length contraction: L′ = L/γ. Divide the proper length by γ.
Key Vocabulary
Centripetal force
The net inward force required to keep an object moving in a circular path. It always points toward the center of the circle.
Example: Fc = mv²/r — for a 2 kg ball at 3 m/s on a 1 m string: Fc = 18 N
Gravitational field
A region of space in which a mass experiences a gravitational force. Field strength g = Fg/m (units: N/kg or m/s²).
Example: At Earth's surface, g ≈ 9.8 N/kg directed toward Earth's center.
Momentum
The product of an object's mass and velocity (p = mv). A vector quantity conserved in closed systems.
Example: A 5 kg object at 10 m/s has p = 50 kg·m/s.
Impulse
The product of a net force and the time interval over which it acts (J = FΔt). Equal to the change in momentum.
Example: A 20 N force acting for 3 s delivers J = 60 N·s of impulse.
Work
Energy transferred by a force acting through a displacement. W = Fd cosθ, where θ is the angle between force and displacement.
Example: Pushing a box 5 m with a 10 N force parallel to the floor: W = 50 J.
Kinetic energy
Energy an object possesses due to its motion. KE = ½mv². Doubles when speed increases by √2; quadruples when speed doubles.
Example: A 4 kg ball at 6 m/s: KE = ½(4)(36) = 72 J.
Potential energy
Stored energy due to an object's position in a force field. Gravitational PE = mgh, measured relative to a chosen reference level.
Example: A 3 kg book on a 2 m shelf: PE = (3)(9.8)(2) ≈ 58.8 J.
Time dilation
The relativistic effect by which a moving clock ticks more slowly than a stationary one. Described by t′ = γt₀, where γ ≥ 1.
Example: At v = 0.87c, γ ≈ 2, so 1 s of proper time appears as 2 s to a stationary observer.
Workbook Check
Interactive Practice — 5 Questions
A 0.8 kg ball moves in a circle of radius 0.5 m at 4 m/s. What is the centripetal force?
If the distance between two masses is doubled, the gravitational force between them becomes:
A 4 kg object moving at 5 m/s collides with and sticks to a stationary 6 kg object. What is their combined speed after the collision?
A 5 kg object is lifted 3 m. How much gravitational potential energy does it gain? (g = 10 m/s²)
A rocket travels at 0.6c. Its proper length is 100 m. What length does a stationary observer measure? (γ = 1.25)