AP Physics 1 - Kinematics and Dynamics Review

Significant Figures and Conversions

• AP Physics 1 exam ignores significant figures; use roughly 3 sig figs.

• Conversions are important for understanding physics.

  • Example: Convert 8,765 kg/m3 to g/cm3.

Vectors and Scalars

• Vectors have magnitude and direction.

• Scalars have magnitude only.

• Examples of vectors:

  • Displacement

  • Velocity

  • Acceleration

  • Force

  • Momentum

  • Torque

  • Angular momentum

• Examples of scalars:

  • Time

  • Distance

  • Mass

  • Speed

  • Volume

  • Density

  • Work

  • Energy

  • Rotational inertia

• Ways to illustrate a vector (e.g., velocity, v):

  • Arrow over the variable: \vec{v}

  • Subscript: v_x

  • Boldface: v

• Vectors are often illustrated using arrows; longer arrows indicate larger magnitudes.

Displacement, Velocity, and Acceleration

• Displacement:

  • Straight-line distance between initial and final points.

  • Change in position.

  • If an object stops where it started, displacement is zero.

  • Distance traveled is always greater than or equal to the magnitude of displacement.

  • Displacement is a vector.

• Average speed:

  • Distance traveled over the time duration of that travel.

  • Equation: \text{Average Speed} = \frac{\text{Distance}}{\text{Time Duration}}

  • Time in the equation is the time duration during which the distance was traveled.

  • Speed is a scalar.

• Average velocity:

  • Equation: \text{Average Velocity} = \frac{\text{Displacement}}{\text{Time Duration}}

  • Typical units: m/s, km/hr, mi/hr, furlongs per fortnight.

  • Velocity is a vector.

  • Instantaneous velocity: Velocity at a specific time (very small time interval).

• Average acceleration:

  • Equation: \text{Average Acceleration} = \frac{\text{Change in Velocity}}{\text{Time Duration}}

  • Typical units: m/s². Only units typically used for linear acceleration.

  • Instantaneous acceleration: Acceleration at a specific time (very small time interval).

  • Acceleration is a vector.

• Uniformly Accelerated Motion (UAM) Equations (Kinematics Equations):

  • Valid when acceleration is constant.

  • Equations on the equation sheet:

    • vf = vi + a\Delta t

    • \Delta x = v_i \Delta t + \frac{1}{2} a \Delta t^2

    • vf^2 = vi^2 + 2 a \Delta x

  • Equation not on the equation sheet:

    • \Delta x = \frac{1}{2} (vf + vi) \Delta t

  • All equations assume initial time is zero.

• Free Fall Motion:

  • Acceleration in the y-direction due to gravity near Earth's surface is approximately 9.81 m/s² down.

  • Can approximate as 10 m/s².

  • Uniformly Accelerated Motion.

  • Velocity at the top of the path in the y-direction is zero.

Motion Graphs

• The slope of a position vs. time graph is velocity.

• The slope of a velocity vs. time graph is acceleration.

• The area between the curve and the time axis on a velocity vs. time graph is change in position.

• The area between the curve and the time axis on an acceleration vs. time graph is change in velocity.

• Area above the horizontal axis is positive; area below is negative.

Two-Dimensional Motion

• Resolve vectors into component vectors.

  • Be careful; theta will not always be with the horizontal, therefore, the x-component will not always use cosine.

• Projectile motion:

  • Motion in two dimensions near a planet's surface where the only force is gravity.

  • Separate known values into x and y-directions:

    • x-direction:

      • a_x = 0

      • Constant velocity

    • y-direction:

      • a_y = -g = -9.81 \frac{m}{s^2} \approx -10 \frac{m}{s^2}

      • Uniformly Accelerated Motion

    • \Delta t is the same in both directions because it is a scalar.

  • Break initial velocity into components; unless the initial velocity is completely horizontal, then v_{iy} = 0.

  • Velocity at the top in the y-direction is zero.

  • When \Delta y = 0, \Delta t{up} = \Delta t{down} and v{iy} = -v{fy}.

Relative Motion

• Motion description changes based on the observer's frame of reference.

• Combine motion of an object and observer using vector addition.

• AP Physics 1 restricts relative motion problems to one-dimensional motion.

Projectile Motion - AP Physics 1: Kinematics Review Supplement

Kinematics serves as the backbone of the AP Physics 1 curriculum.

• understanding motion graphs

• working with multiple variables and multiple equations

• breaking vectors into components.

Problem 1

A projectile is launched with speed vi at an angle of θ above the horizontal. At its maximum height, the horizontal and vertical components of the projectile’s velocity and acceleration are:

• Horizontal Velocity Component v_i cos \theta

• Vertical Velocity Component 0

• Vertical Acceleration Component -g

Problem 2

A small steel ball rolls off a horizontal table with a height of 1.2 m and lands a horizontal distance of 0.55 m from the edge of the table. What was the speed of the ball as it rolled on the table?

• The correct answer is (D) 1.1 m/s

Problem 2a

A small steel ball rolls off a horizontal table with a height of h and lands a horizontal distance of d from the edge of the table. What was the speed of the ball as it rolled on the table?

• The correct answer is still (D).

Motion Graphs - AP Physics 1: Kinematics Review Supplement

Problem 1

You slam your foot down on the accelerator pedal in your car causing it to speed up with a uniform acceleration. After a few seconds, you take your foot off the accelerator pedal and immediately slam it down on the brake pedal, causing your car to slow down with a uniform acceleration. Your velocity as a function of time graph is shown. Which graph could correctly show your position as a function of time?

• The key to remember here is that the slope of a position vs. time graph is velocity.

• The correct answer is (A).

Problem 2

The graph shows the acceleration of a particle with respect to time. Assuming the velocity of the particle at t = 0 seconds is - 10 m/s, which of the following is the velocity of the particle at t = 8 seconds?

• The area “under” an acceleration as a function of time graph is change in velocity, however, remember that area under the time axis is negative and area above the time axis is positive.

• The correct answer is (A) -2 m/s

Motion Graphs Concepts

• the slope of a position versus time graph is velocity,

• the slope of a velocity versus time graph is acceleration,

• the area1 “under” a velocity versus time graph is change in position,

• the area “under” an acceleration versus time graph is change in velocity

Center of Mass

• Equation on the equation sheet:

  • x{cm} = \frac{\sumi mi xi}{\sumi mi}

    • Expressed alternatively:

      • x{cm} = \frac{m1x1 + m2x2 + m3x3 + …}{m1 + m2 + m3 + …}

    • x can be y or z depending on the direction.

    • Position is relative to a zero point which could be the origin or elsewhere.

• Can also refer to velocity and acceleration of the center of mass:

  • v{cm} = \frac{\sumi mi vi}{\sumi mi}

  • a{cm} = \frac{\sumi mi ai}{\sumi mi}

Center of Mass of an Object with Shape

• Object: a collection of particles with little to no interaction.

• Object: treated as having no internal structure.

• Problems in AP Physics 1 should only include:

  • Uniform density objects with obvious centers of mass.

  • Centers of mass that can be estimated.

• Examples:

  • Center of mass of a uniform density sphere is at its center.

  • Center of mass of a uniform density rectangular box is at its center.

Forces

• All forces are vectors.

• All forces result from interaction between two objects. Always!

  • An object cannot exert a force on itself!

• Free Body Diagrams show all forces acting on an object.

  • Only force vectors should be in free body diagrams.

  • Do not include: displacement, velocity, acceleration, momentum, etc.

  • All forces start at the center of mass of the object or system.

  • If two or more forces act in the same direction, they still start at the center of mass but are offset.

  • Never break forces into components in a free body diagram on an AP Physics exam; redraw the diagram elsewhere.

  • Five steps to solve any free body diagram problem:

    1. Draw the free body diagram.

    2. Break forces into components.

    3. Redraw the free body diagram.

    4. Sum the forces.

    5. Sum the forces in a direction perpendicular to the direction in step 4.

• Force normal:

  • Caused by a surface, always perpendicular to the surface, and pushes away from that surface.

• Force of tension:

  • In a rope, string, cable, chain, or similar.

  • If the rope is ideal, it has negligible mass and does not stretch.

  • Tension has the same magnitude at all points in an ideal rope.

    • If the rope does not have negligible mass, the force of tension may not be the same at all points in the rope.

  • Tension is always parallel to the direction of the rope, wire, string, or cable.

• Contact forces:

  • Result from the interaction of one object touching another object and result from electric forces between the atoms of the objects.

  • Examples:

    • Force of tension

    • Force of friction

    • Force normal

    • Force applied

    • Spring force

Newton’s First Law

• “An object at rest will remain at rest and an object in motion will remain at a constant velocity unless acted upon by a net, external force.”

  • Objects in motion will remain at a constant velocity.

  • Unless acted upon by “a net, external force”.

• Law of Inertia.

  • Inertia: tendency of an object to resist a change in state of motion.

  • Inertia: tendency of an object to resist acceleration.

• Valid only when measurements are taken from an inertial reference frame.

  • Acceleration of an inertial reference frame is zero.

  • On the AP Physics 1 exam, all reference frames are assumed to be inertial, unless otherwise stated.

Newton’s Second Law

• On the AP Physics equation sheet: \vec{a} = \frac{\vec{F}_{net}}{m}

• How I usually use it: \vec{F}_{net} = m\vec{a}

  • They are the same equation rearranged.

• Units: newtons (N).

  • 1 N = 1 \frac{kg \cdot m}{s^2}

• Identify the object or system on which we are summing the forces and the direction in which we are summing the forces.

• Acceleration is always in the same direction as the net force.

• Translational equilibrium:

  • Net force on the object equals zero.

  • Object is either at rest or moving at a constant velocity because the acceleration of the object equals zero.

  • \sum \vec{F} = 0

  • If an object is at rest, there could be forces acting on the object from other objects, however, the net force acting on the object is zero.

Newton’s Third Law

• For every force object one exerts on object two, object two exerts an equal but opposite force on object one.

  • \vec{F}{1on2} = -\vec{F}{2on1}

  • These two forces act simultaneously.

• Forces internal to a system do not change the motion of the center of mass of the system.

Gravitational Force

• Interaction between an object with mass and another object with mass.

  • “Gravitational force”, “force of gravity”, and “weight” mean the same thing.

  • The gravitational force is always between two objects.

• Magnitude of the gravitational force exerted on a mass in a gravitational field:

  • F_g = mg

  • In the same direction as the gravitational field.

    • In Free Fall a_y = -g

  • Direction of the force of gravity: towards the center of mass of the planet; down.

  • The two objects interacting: the object and the planet causing the gravitational field.

• Inertial mass: measure of an object's inertia or resistance to acceleration.

  • Mass in Newton’s Second Law: \vec{F}_{net} = m\vec{a}

• Gravitational mass: mass used to determine the force of gravity, or weight, of an object.

  • Mass in the gravitational force equation: F_g = mg

• Inertial mass and gravitational mass are mathematically equivalent.

  • This has been experimentally verified.

Force of Friction

• Direction:

  • Parallel to the surface.

  • Always opposes sliding motion.

  • Independent of the direction of the force applied.

• Equation:

  • F*f = \mu F_N

    • Coefficient of friction, μ:

      • \mu = \frac{Ff}{FN}

        • Ratio of the maximum force of friction and the force normal.

        • No units.

        • Cannot be negative.

        • Typical value are between and 2.

        • Experimentally determined.

      • Kinetic Friction:

        • F{fk} = \muk F_N

          • Two surfaces are sliding relative to one another.

      • Static Friction:

        • F{fs} \le \mus F_N

          • Two surfaces are not sliding relative to one another.

          • Adjusts to prevent sliding.

          • Maximum force:

            • F{fsmax} = \mus F_N

    • For two surfaces, the coefficient of static friction is almost always more that the coefficient of kinetic friction.

      • Takes more force to put an object into motion than it takes to keep an object moving.

• Friction does not depend on the size of the surface area of contact.

Force on Objects on Inclines

• Break the force of gravity into components parallel and perpendicular to the incline before summing forces.

• Force of gravity perpendicular:

  • F_{g\perp} = mg \cos(\theta)

• Force of gravity parallel:

  • F_{g||} = mg \sin(\theta)

  • Directed down the incline.

Center of Mass - AP Physics 1: Kinematics Review Supplement

Problem 1

A giant, spherical helium balloon with mass M and radius R has a massless rope of length 3R hanging from it. A person with a mass of M is hanging on the rope in perfect equilibrium such that the balloon-person system does not move up or down. If the person starts a distance R from the bottom of the balloon and climbs down to the end of the rope, which figure best illustrates the final position of the balloon and person? Assume there is no wind.

• Because the net external force on the balloon-person system is zero, the system will not accelerate, and the center of mass of the system will stay in the same location while the person climbs down to the bottom of the rope.

• Therefore, the correct answer is (B).

Problem 2

While peacefully reading your physics textbook, you are sliding at 3 m/s East on a very large patch of frozen, frictionless ice. In frustration, you decided to throw your physics textbook and give it a speed of 8 m/s North. If your mass is 40 times larger than the mass of your physics textbook, what is the velocity of the center of mass of the you-textbook system after you throw the book?

• The correct answer is (B) 3 m/s East

Newton's Second Law - AP Physics 1: Dynamics Review Supplement

Problem 1

A rock is freely falling downward toward the Earth. The weight of the rock is W. Which of the following best describes the force from the rock on the Earth as the rock falls?

• The correct answer is (B) W upward

Problem 2

A 300 N object is accelerating to the North at 1 m/s2. A force, F1, of 40 N acts due East on the object. If there is only one other force, F2, acting on the object, what is the magnitude and direction of F2?

• The correct answer is (A) 50 N @ 37° N of W

Problem 3

Which of the following two free body diagrams could be for an object which has a constant velocity in the y-direction and a constant acceleration in the x-direction?

• Choice A and D are correct answer.

Friction - AP Physics 1: Dynamics Review Supplement

Problem 1

As shown, a block of mass mb on a table is attached to a string which goes over an ideal pulley and is attached to a cube of mass mc. The block and cube are currently at rest. If the coefficient of static friction between the block and the table is μs, the magnitude of the force preventing the block from accelerating is best described by:

• The correct answer is D. mcg

Problem 2

A book with mass m is at rest on an incline of angle θ as shown. If the coefficient of static friction between the book and the incline is μs, and the coefficient of kinetic friction between the book and the incline is μk, which expression best represents the force of friction currently acting on the book?

• The correct answer is D. mgsinθ

Problem 3

A 20 kg mass is at rest on a table. A horizontal force of 140 N is applied to a block, however, it is not enough to move the block. Which of the following best describes what we know about the coefficient of static friction between the block and the table?

• The correct answer is A. μs ≥ 0.7

Universal Gravitation and Gravitational Field

• Newton’s Law of Universal Gravitation:

  • Fg = G \frac{m1 m_2}{r^2}

    • G: Gravitational Constant (on the Table of Information).

    • m: gravitational masses.

    • r: distance between the centers of mass of the two objects.

    • Force is directed along a line connecting the centers of mass.

    • Forces acting on both masses have the same magnitude and form a Newton’s Third Law force pair.

• Local gravitational field, g:

  • Humans live in a narrow band of altitudes where g is nearly constant and directed downward.

    • g = \frac{F_g}{m}

    • Rearrangement of the gravitational force equation: F_g = mg

    • Describes the interaction between two masses: the mass of the object and the mass of the planet creating the gravitational field.

• Derivation of the gravitational field, g, on the surface of a planet:

  • g = G \frac{M}{r^2}

• If the only force acting on the object is the gravitational force, then the object is in free fall, and , and the acceleration of the object has the same magnitude as the gravitational field, g.

• Units:

  • Free fall acceleration: m/s²

  • Gravitational field: N/kg

  • These units are equivalent: \frac{N}{kg} = \frac{m}{s^2}

• Approximation of g near the surface of Earth:

  • g \approx 10 \frac{m}{s^2}

Spring Force

• Ideal spring force is proportional to its displacement from equilibrium position.

• Hooke's Law:

  • F_s = -k \Delta x

    • k: spring constant (always positive).

    • Δx: displacement from equilibrium.

    • The negative sign indicates the spring force and the displacement are opposite in direction.

• Ideal spring has negligible mass.

• The magnitude of the slope of a graph of spring force vs. displacement from equilibrium position is the spring constant.

Circular Motion

• Tangential velocity:

  • Linear velocity of an object moving along a circular path.

  • Directed perpendicularly to the radius and parallel to the path.

• Centripetal acceleration:

  • Acceleration directed inward toward the center of the circle.

  • An object must have centripetal acceleration because the direction of tangential velocity is always changing.

  • a_c = \frac{v^2}{r}

• Period (T):

  • Time it takes to complete one circle.

• Frequency (f):

  • Number of revolutions per second.

  • Cycles per second or hertz (Hz).

  • f = \frac{1}{T}

• Period equation:

  • v = \frac{2 \pi r}{T}

• Centripetal force:

  • Net force in the in direction or the