Instantaneous Velocity and Speed

• Motion universally occurs, from walking to the movement of galaxies.
• Motion represents a change in an object's position relative to time.
• The chapter addresses describing motion.
• The chapter will cover velocity, also acceleration, and rectilinear motion.
• Relative velocity explains motion’s relative nature.
• Objects are considered as points if their size is small relative to the distance they travel.
• Kinematics describes movement, and Chapter 4 addresses the causes of movement.

Instantaneous Velocity and Speed

• Instantaneous velocity, "v", is an object's velocity at a specific time, defined as the limit of average velocity when the time interval approaches zero.
• Instantaneous velocity equation: v = lim (Δx/Δt) as Δt approaches 0
• Expressed as a derivative: v = dx/dt
• The instantaneous velocity at a point can be found graphically by determining the slope of a tangent line on a position-time graph, and doing so numerically.
• Instantaneous speed refers to the magnitude of velocity.
• +24.0 m/s and -24.0 m/s have speeds of 24.0 m/s.
• Average speed is equal to or greater than average velocity magnitude, but instantaneous speed equals instantaneous velocity magnitude.

Acceleration

• Acceleration describes how velocity changes during motion.
• Galileo found that the rate of velocity change is constant over time for free-falling objects, leading to acceleration’s concept.
• Average acceleration relates to velocity change over a time interval.
• Average acceleration formula: a = (v2 - v1) / (t2 - t1) = Δv / Δt
• SI unit for acceleration: m/s².
• Average acceleration on a velocity-versus-time plot can be determined by the slope of a line that connects points (v2, t2) and (v1, t1).
• Instantaneous acceleration is the limit of average acceleration when Δt approaches zero: a = lim (Δv/Δt) as Δt→0 = dv/dt.
• Acceleration is the slope on a v-t curve at any given time.
• Velocity changes in magnitude or direction create acceleration.
• Like velocity, acceleration may have positive, negative, and zero values.
• In position-time graphs, positive acceleration curves upwards, and negative curves downwards, while zero acceleration is a straight line.
• Average acceleration matches constant acceleration.
• If velocity starts at v₀ at t=0 and is v at time t, then a = (v - v₀) / (t - 0), so v = v₀ + at.
• In the case of any moving object, a velocity-time graph shows its area illustrates displacement in the given time.

Kinematic Equations for Uniformly Accelerated Motion

• For uniform acceleration, the following variables relate: displacement (x), time (t), initial velocity (v₀), final velocity (v), and acceleration (a).
• Equation that relates final and initial velocities: v = v₀ + at
• Displacement: x= v₀*t + (½)at²
• Displacement can be written as x = ((v + v₀)/2) * t = v*t
• The object's average velocity equals the arithmetic average of the final and initial velocities.
• From equation v = v₀ + at then t = (v - v₀) / a
• With substitution then x = ((v + v₀)/2) / ((v - v₀) / a) becomes v² = v₀² + 2ax

Points to Ponder

• Axis origin and direction choice matters
• Speeding up means acceleration is then in the direction of speed, while slowing is in the opposite direction.
• Sign of acceleration does not indicate speeding up/ slowing down
• Zero velocity doesn't mean zero acceleration
• Quantities in motion equations are algebraic, applied with correct signs
• Instantaneous velocity and acceleration are always correct, while kinematic equations are only correct if acceleration is constant.