Unit 1 Beta

Kinematics: the one-page sheet

1.1 Scalars and Vectors on a Line

Scalars have a size only; vectors have a size and a direction, which along a line is the sign. Displacement is the final minus the starting position; distance is the path length. Speed is the magnitude of velocity, and vectors on a line add as signed numbers.

A scalar has a size only (distance, speed, time). A vector has a size and a direction (position, displacement, velocity).
Choose a coordinate system: an origin and a positive direction. Then a position is one signed number, x.
Displacement = final position − starting position. Distance = the total length of the path. They match only for a trip in one direction.
Speed is how fast; velocity is how fast and which way. Along a line the sign of the velocity is the direction of motion.
Add vectors on a line by adding signed values: +320 m and −485 m give −165 m.

every position along the line becomes one signed number, x, in meters its displacement is the final position minus the starting position, with a sign for direction the distance keeps growing, but parts of the displacement cancel, so the distance is larger their vector sum is found by adding the signed values

scalar: A quantity with a size (and a unit) but no direction, such as distance, speed, time or mass.
vector: A quantity with both a size and a direction, such as displacement or velocity. Along a line, its sign gives its direction.
magnitude: The size of a vector, with no direction: always zero or positive. The magnitude of a velocity of −4.2 m/s is 4.2 m/s.
position: Where an object is, measured from an origin along a chosen positive direction, in meters. Along a line it is one signed number, x.
coordinate system: The choice of an origin (zero point) and a positive direction (an x-axis) used to give positions as numbers.
distance: The total length of the path an object travels: a scalar that counts every meter and never decreases.
displacement: The change in position: final position minus starting position. A vector; along a line its sign gives its direction.
speed: How fast an object moves, as a distance per time (m/s): a scalar, the magnitude of the velocity.
velocity: How fast an object moves and in which direction: displacement per time, in m/s. A vector; its magnitude is the speed.
vector sum: Adding vectors along one line by adding their signed values: +12 m and −5 m add to +7 m. The result is the resultant.

1.2 Displacement, Velocity and Acceleration

Average velocity is displacement over time, Δx/Δt, and average acceleration is change in velocity over time, Δv/Δt. Acceleration is any change in velocity and points along Δv: with v, the object speeds up; against v, it slows down. Instantaneous values are averages over very short intervals.

The object model treats a car or a ball as a point with a position: size, shape and inner parts are ignored.
Average velocity = Δx/Δt. It uses the displacement, so a round trip has zero average velocity. Average speed uses the distance.
Average acceleration = Δv/Δt, in m/s². Acceleration is any change in velocity: speeding up, slowing down or turning.
The sign of a alone does not tell you whether an object speeds up. Compare it with v: same sign, speeding up; opposite signs, slowing down.
An instantaneous value is the average over a very short interval. An object can have v = 0 at an instant and still be accelerating.

its average velocity is Δx/Δt, with the sign of the displacement the average velocity approaches the instantaneous velocity at that moment the object has an acceleration, a = Δv/Δt, pointing the same way as Δv the object speeds up; if it points the opposite way, the object slows down

object model: The object model: treating an object as a single point that has a position but no size, shape or inner parts. It works when those do not affect the motion described.
time interval: The time between two moments: Δt = t(final) − t(initial), in seconds.
average velocity: Displacement divided by the time interval: v(avg) = Δx/Δt, in m/s. Its sign gives the direction of the net change in position.
acceleration: The rate at which velocity changes, in m/s². An object accelerates when its speed changes, its direction changes, or both.
average acceleration: Change in velocity divided by the time interval: a(avg) = Δv/Δt, in m/s². It points the same way as Δv.
instantaneous velocity: The velocity at one instant: the value the average velocity approaches as the time interval is made very short.
instantaneous acceleration: The acceleration at one instant: the value the average acceleration approaches as the time interval is made very short.

1.3 Representing Motion

Motion can be shown as a motion diagram, x-t, v-t and a-t graphs, equations and words. Slopes go down the chain (x-t to v-t to a-t) and areas go up it. For constant acceleration, three kinematic equations apply, each missing one quantity; free fall is constant acceleration g downward.

A motion diagram shows positions at equal times: growing gaps mean speeding up, shrinking gaps mean slowing down.
Slope of x-t = velocity. Slope of v-t = acceleration. Area under v-t = displacement. Area under a-t = change in velocity.
For constant acceleration: v = v₀ + at; x = x₀ + v₀t + ½at²; v² = v₀² + 2a(x − x₀). Pick the one that leaves out the quantity you neither know nor want.
Free fall: a = g = 9.8 m/s² downward (10 m/s² is accepted), on the way up, at the top and on the way down.
If the acceleration changes (nonuniform), the v-t graph curves. Describe it from the graph only; the equations do not apply.

the slope of the x-t graph at any instant is the velocity the slope of the v-t graph is the acceleration, and the area under it is the displacement the v-t graph is a straight line and the three kinematic equations apply its acceleration is g = 9.8 m/s² downward at every instant, even when v = 0 at the top

motion diagram: A picture of an object's position at equal time intervals, as dots or images, often with velocity arrows. Growing gaps mean speeding up; shrinking gaps, slowing down.
position-time graph: A graph of position x against time t. Its slope at any instant is the velocity; a straight line means constant velocity, a curve means the velocity changes.
velocity-time graph: A graph of velocity v against time t. Its slope is the acceleration and the area between it and the time axis is the displacement.
acceleration-time graph: A graph of acceleration a against time t. The area between it and the time axis is the change in velocity.
constant acceleration: Motion in which the velocity changes by the same amount in every equal time interval: a straight, sloped v-t graph and a horizontal a-t graph.
kinematic equations: The three equations for constant acceleration: v = v₀ + at; x = x₀ + v₀t + ½at²; v² = v₀² + 2a(x − x₀). Each leaves out one quantity.
free fall: Motion under gravity alone, with air resistance negligible. Near Earth every freely falling object has a constant acceleration g = 9.8 m/s² (about 10 m/s²) downward.
nonuniform acceleration: Acceleration that changes with time: a curved v-t graph. In this course it is described from graphs and in words, not calculated.

1.4 Reference Frames and Relative Motion

Every velocity is measured relative to a reference frame. Along a line, a velocity relative to the ground is the velocity relative to a moving frame plus that frame's velocity; swapping observers flips the sign. Frames moving at constant velocity relative to each other agree on accelerations.

A reference frame is the observer's point of view, with a coordinate system that moves with the observer. "At rest" means at rest relative to some frame.
Relative velocity along a line: v(object, ground) = v(object, frame) + v(frame, ground). Give each velocity its sign.
To find a velocity relative to a moving observer, subtract the observer's velocity: v(A, B) = v(A, ground) − v(B, ground).
Frames moving at a constant velocity relative to each other measure the same acceleration, because the added velocity cancels in Δv.

they measure different velocities for the same object its velocity relative to the ground is v + V (signed, along one line) the relative velocity flips sign: if B moves at +3 m/s relative to A, A moves at −3 m/s relative to B the same velocity is added at every instant, so both measure the same acceleration

reference frame: The point of view from which motion is measured: an observer with a coordinate system that moves with them. A velocity has meaning only relative to some frame.
relative velocity: The velocity of an object as measured in a given frame. Along a line: v(object relative to ground) = v(object relative to frame) + v(frame relative to ground).

1.5 Vectors and Motion in Two Dimensions

Two-dimensional motion is two independent one-dimensional motions sharing the same time. A projectile keeps a constant horizontal velocity v cos θ and falls freely with vertical acceleration g; the vertical motion sets the time of flight, and the range is the horizontal velocity times that time.

Treat two-dimensional motion as two one-dimensional motions at right angles. They share only the time.
A projectile: aₓ = 0, so vₓ is constant; a_y = g downward, so the vertical motion is free fall.
Split the launch velocity: vₓ = v cos θ, v_y = v sin θ (θ, the launch angle, measured from the horizontal).
The time of flight comes from the vertical motion only. The horizontal range = vₓ × time of flight.
At the top, v_y = 0 but vₓ is unchanged and the acceleration is still g downward.

it splits into a horizontal component v cos θ and a vertical component v sin θ the horizontal acceleration is zero and the vertical acceleration is g downward you solve each as a one-dimensional problem, linked only by the shared time the range is the constant horizontal velocity times that time of flight

independence of perpendicular motions: In two-dimensional motion, the horizontal and vertical components each follow their own one-dimensional rules; what happens in one direction does not affect the other. Only the time is shared.
projectile: An object that, after launch, moves under gravity alone (air resistance negligible): zero horizontal acceleration and a constant downward acceleration g.
launch angle: The angle between a projectile's initial velocity and the horizontal. It sets how the launch speed splits: horizontal v cos θ, vertical v sin θ.
horizontal range: The horizontal distance a projectile travels before landing: the horizontal velocity times the time of flight.
time of flight: How long a projectile is in the air. It is set by the vertical motion alone: the vertical launch velocity, the height and g.