What you'll learn
Motion and Forces forms a foundational unit in US Common Core Physics, examining how objects move and the forces that cause or resist that movement. This topic accounts for approximately 20-25% of most state standardized physics assessments and provides essential understanding for kinematics, dynamics, and energy concepts tested throughout the course.
Key terms and definitions
Velocity — the rate of change of displacement; a vector quantity measured in meters per second (m/s) that includes both speed and direction.
Acceleration — the rate of change of velocity; measured in meters per second squared (m/s²), calculated as change in velocity divided by time taken.
Net force — the vector sum of all forces acting on an object; determines the object's acceleration according to Newton's Second Law.
Friction — a contact force that opposes motion between two surfaces in contact; depends on the nature of the surfaces and the normal force pressing them together.
Momentum — the product of an object's mass and velocity (p = mv); a vector quantity measured in kilogram-meters per second (kg·m/s).
Inertia — the tendency of an object to resist changes in its state of motion; directly proportional to the object's mass.
Normal force — the perpendicular contact force exerted by a surface on an object resting on or pressed against it.
Newton — the SI unit of force (N); one newton equals the force required to accelerate a 1 kg mass at 1 m/s².
Core concepts
Newton's First Law of Motion
Newton's First Law states that an object at rest stays at rest, and an object in motion continues moving at constant velocity, unless acted upon by a net external force. This principle, also called the law of inertia, directly contradicts everyday intuition because friction is nearly always present in real-world scenarios.
Key exam applications:
- Objects on frictionless surfaces continue moving indefinitely at constant velocity
- When balanced forces act on an object (net force = 0), acceleration equals zero
- Passengers lurch forward during sudden braking because their bodies tend to maintain constant velocity
- Greater mass means greater resistance to changes in motion
Free-body diagrams are essential for First Law problems. Every force acting on an object must be represented as an arrow, with length proportional to magnitude and direction showing the force's orientation. When forces balance, arrows in opposite directions have equal length.
Newton's Second Law of Motion
Newton's Second Law quantifies the relationship between force, mass, and acceleration through the equation:
F = ma
where F is net force (N), m is mass (kg), and a is acceleration (m/s²).
Critical exam points:
- The force in F = ma represents the net force (sum of all forces), not individual forces
- Acceleration occurs in the same direction as the net force
- Doubling the net force doubles the acceleration for constant mass
- Doubling the mass halves the acceleration for constant net force
- This relationship is linear and proportional
Common exam scenarios include:
- Calculating the force needed to accelerate a car of known mass
- Determining acceleration when multiple forces act on an object
- Finding an object's mass from its acceleration under known force
- Analyzing elevator motion with changing normal forces
Newton's Third Law of Motion
Newton's Third Law states that for every action force, there exists an equal and opposite reaction force. These force pairs:
- Always act on different objects (never the same object)
- Have equal magnitudes
- Point in opposite directions
- Occur simultaneously (not sequentially)
- Are the same type of force (both gravitational, both contact, etc.)
Exam-tested examples:
- A swimmer pushes water backward (action); water pushes swimmer forward (reaction)
- Earth pulls down on a book with gravitational force; book pulls up on Earth with equal force
- A rocket expels exhaust gases downward (action); gases push rocket upward (reaction)
- Your foot pushes on the ground backward when walking; ground pushes your foot forward
The most common error is identifying action-reaction pairs on the same object. Forces balancing on a single object are not action-reaction pairs.
Friction forces
Friction opposes relative motion between surfaces. Two types appear on US Common Core assessments:
Static friction prevents surfaces from sliding past each other; it increases to match applied force up to a maximum value:
f_s ≤ μ_s N
where μ_s is the coefficient of static friction and N is the normal force.
Kinetic friction opposes surfaces already sliding past each other; it has constant magnitude:
f_k = μ_k N
where μ_k is the coefficient of kinetic friction.
Key relationships:
- Static friction coefficient (μ_s) exceeds kinetic friction coefficient (μ_k) for the same surfaces
- Greater normal force produces greater friction
- Friction acts parallel to the contact surface
- Friction is independent of contact area (counterintuitive but testable)
- Friction converts kinetic energy to thermal energy
Exam questions often require calculating the force needed to overcome static friction or the deceleration caused by kinetic friction.
Kinematics equations
For objects moving with constant acceleration, four kinematic equations relate displacement (Δx), initial velocity (v₀), final velocity (v), acceleration (a), and time (t):
- v = v₀ + at
- Δx = v₀t + ½at²
- v² = v₀² + 2aΔx
- Δx = ½(v₀ + v)t
Equation selection strategy:
- Identify which variable is unknown
- Identify which variable is not given and not needed
- Select the equation that contains the unknown but not the unnecessary variable
- Ensure consistent units (convert if necessary)
These equations apply only when acceleration remains constant. Variable acceleration requires calculus-based approaches beyond Common Core scope.
Motion graphs
Position-time graphs:
- Slope equals velocity
- Horizontal line indicates zero velocity (rest)
- Straight diagonal line indicates constant velocity
- Curved line indicates changing velocity (acceleration)
- Steeper slope indicates greater speed
Velocity-time graphs:
- Slope equals acceleration
- Horizontal line indicates constant velocity (zero acceleration)
- Area under the curve equals displacement
- Positive slope indicates increasing velocity
- Negative slope indicates decreasing velocity or increasing speed in negative direction
Acceleration-time graphs:
- Area under the curve equals change in velocity
- Horizontal line indicates constant acceleration
Graph interpretation questions constitute 15-20% of motion problems on standardized assessments. Practice converting between graph types and extracting quantitative information from slopes and areas.
Momentum and impulse
Momentum (p = mv) measures the "quantity of motion" an object possesses. The law of conservation of momentum states that in a closed system with no external forces, total momentum before a collision equals total momentum after:
m₁v₁ᵢ + m₂v₂ᵢ = m₁v₁f + m₂v₂f
This principle applies to:
- Elastic collisions (kinetic energy conserved)
- Inelastic collisions (kinetic energy not conserved)
- Explosions and separations
- Multi-dimensional collisions (apply separately to x and y components)
Impulse (J) equals the change in momentum:
J = Δp = FΔt
This impulse-momentum theorem explains why:
- Airbags reduce injury by increasing collision time (reducing force)
- Following through in sports maximizes force application time
- Landing with bent knees reduces impact force
Exam problems often require calculating final velocities after collisions or determining forces from momentum changes.
Worked examples
Example 1: Newton's Second Law application
Question: A 1,200 kg car accelerates from rest to 25 m/s in 8.0 seconds on a straight road. Calculate (a) the acceleration and (b) the net force acting on the car.
Solution:
(a) Using v = v₀ + at:
- v₀ = 0 m/s (starts from rest)
- v = 25 m/s
- t = 8.0 s
- a = (v - v₀)/t = (25 - 0)/8.0 = 3.125 m/s²
- Rounded appropriately: a = 3.1 m/s²
(b) Using F = ma:
- m = 1,200 kg
- a = 3.125 m/s²
- F = 1,200 × 3.125 = 3,750 N
- F = 3,800 N (rounded to 2 significant figures)
Mark scheme: Part (a) worth 2 marks — 1 for correct equation, 1 for answer with units. Part (b) worth 2 marks — 1 for using F = ma correctly, 1 for final answer.
Example 2: Friction problem
Question: A 50 kg wooden crate rests on a horizontal floor. The coefficient of static friction is 0.40 and the coefficient of kinetic friction is 0.30. What minimum horizontal force is required to start the crate moving?
Solution:
First, find the normal force:
- N = mg = 50 × 9.8 = 490 N (weight on horizontal surface)
The maximum static friction before motion begins:
- f_s(max) = μ_s N = 0.40 × 490 = 196 N
The minimum force to overcome static friction:
- F = 196 N or 2.0 × 10² N
Once moving, kinetic friction would be:
- f_k = μ_k N = 0.30 × 490 = 147 N (this is not asked but commonly appears as a follow-up)
Mark scheme: 3 marks total — 1 for calculating normal force, 1 for applying f_s = μ_s N, 1 for correct numerical answer with unit.
Example 3: Conservation of momentum
Question: A 0.50 kg ball moving at 4.0 m/s collides head-on with a 0.30 kg ball moving at 2.0 m/s in the opposite direction. After collision, they stick together. Calculate their final velocity.
Solution:
Define positive direction as the direction of the 0.50 kg ball:
- m₁ = 0.50 kg, v₁ᵢ = +4.0 m/s
- m₂ = 0.30 kg, v₂ᵢ = -2.0 m/s (opposite direction)
Total initial momentum:
- pᵢ = m₁v₁ᵢ + m₂v₂ᵢ = (0.50)(4.0) + (0.30)(-2.0)
- pᵢ = 2.0 - 0.60 = 1.4 kg·m/s
After sticking together (perfectly inelastic):
- Combined mass = 0.50 + 0.30 = 0.80 kg
- pf = pᵢ
- (0.80)vf = 1.4
- vf = 1.4/0.80 = 1.75 m/s or 1.8 m/s in the original direction of the 0.50 kg ball
Mark scheme: 4 marks — 1 for recognizing momentum conservation, 1 for correct initial momentum calculation with signs, 1 for correct setup of final equation, 1 for answer with direction.
Common mistakes and how to avoid them
Confusing mass and weight: Mass (kg) measures quantity of matter; weight (N) is the gravitational force (W = mg). Always use mass in F = ma, not weight. Weight only appears when gravity is one of the forces being analyzed.
Forgetting vector nature of forces: Adding forces algebraically without considering direction produces wrong net force. Always establish a coordinate system and assign positive/negative signs based on direction, or use component vectors for 2D problems.
Using the wrong kinematic equation: Students often default to Δx = v₀t + ½at² for all problems. Check which variables are given and which is unknown, then select the equation that connects them without requiring an unknown intermediate value.
Misidentifying action-reaction pairs: Action-reaction forces act on different objects, not the same object. The weight of a book and the normal force from a table are NOT action-reaction pairs; they act on the same book and are different force types (gravitational vs. contact).
Assuming kinetic friction equals maximum static friction: Kinetic friction (μ_k N) is always less than maximum static friction (μ_s N) for the same materials. Once an object starts moving, less force is needed to keep it moving than was needed to start it.
Sign errors in momentum problems: Establish a positive direction at the start. Velocities in the opposite direction get negative signs. Maintain consistent signs throughout the calculation, especially in collision problems involving opposite directions.
Exam technique for Motion and Forces
Command word "Calculate" requires numerical work with units. Always show equation used, substitution of values with units, and final answer. Typical allocation: 2-4 marks depending on steps required.
Command word "Explain" demands physics reasoning, not just description. State the principle or law, then apply it to the specific scenario. For example: "Newton's First Law states objects maintain constant velocity unless acted on by net force. The spacecraft continues moving because space is frictionless, so no net force acts on it."
Free-body diagrams earn marks for: drawing all forces as arrows from the center of the object, labeling each force clearly, showing relative magnitudes through arrow length, and indicating direction accurately. Partial credit available even if subsequent calculations are wrong.
Multi-step problems typically award method marks and answer marks separately. Show all working clearly. Even if the final answer is incorrect due to a calculation error, correct method earns most of the marks. Circle or box final answers for clarity.
Quick revision summary
Newton's First Law: objects maintain constant velocity unless net force acts. Newton's Second Law: F = ma relates net force, mass, and acceleration. Newton's Third Law: action-reaction force pairs are equal, opposite, and act on different objects. Friction opposes motion; static exceeds kinetic. Kinematic equations apply only for constant acceleration. Momentum (p = mv) is conserved in closed systems. Impulse (FΔt) equals momentum change. Graph slopes and areas reveal motion quantities. Master free-body diagrams and equation selection for exam success.