Kinematics: Velocity, Speed, and Acceleration — Study Notes

Velocity, Speed, and Acceleration — Lecture Notes

• Key ideas introduced
  • Average velocity and instantaneous velocity are distinct concepts.
  • Objects have displacement, distance traveled, velocity, speed, and acceleration in various forms; each has a specific meaning and units.
  • Vectors vs. scalars: displacement, velocity, and acceleration are vectors; time and distance traveled are scalars (though distance traveled is a scalar quantity).

Average velocity

• Definition and interpretation
  • Average velocity is the displacement per unit time.
  • It is the magnitude of displacement in one unit of time, with a direction.
  • It is a vector quantity; its direction is the same as the direction of the displacement.
  • If the object moves to the right, the displacement is positive; if it moves to the left, the displacement is negative.
  • The direction of average velocity is the same as the direction of the net displacement.
• Formula and components
  • $\vec{v}<em>{\text{avg}} = \frac{\Delta \vec{x}}{\Delta t} = \frac{\vec{x}</em>{\text{final}} - \vec{x}<em>{\text{initial}}}{t</em>{\text{final}} - t_{\text{initial}}}$
  • \Delta \vec{x} is the displacement vector; \Delta t is the elapsed time (scalar).
  • Magnitude of average velocity is the distance divided by total time only if displacement is along a straight line with no change in direction; more generally, the magnitude is |\vec{v}_{\text{avg}}|.
• Distinguishing displacement vs. distance traveled
  • Displacement (\Delta x) is the net change in position: from initial to final point (a vector).
  • Distance traveled is the total length of the path traveled; it can be greater than the magnitude of displacement.
  • Average velocity uses displacement (not total distance): it divides net change in position by total time.
• Example (Dalton to Atlanta)
  • Suppose displacement from Dalton to Atlanta is 100 miles and the time taken is 1 h 40 min.
  • Convert time: 1 h 40 min = 1 + 40/60 = 5/3 hours.
  • $v_{\text{avg}} = \frac{100\ \text{mi}}{\frac{5}{3}\ \text{h}} = 60\ \text{mi/h}.$
  • The reported average velocity could be 60 mph; actual instantaneous speeds along the trip may be higher or lower depending on segments.
• Practical notes
  • You report an average velocity for the whole trip as displacement over total time, not the average of instantaneous speeds.
  • Direction of \vec{v}_{\text{avg}} aligns with the overall displacement direction.

Instantaneous velocity

• Definition and interpretation
  • Instantaneous velocity is the velocity of the object at a specific instant in time.
  • As the object moves, velocity can change from instant to instant (i.e., velocity is not necessarily constant).
• How it’s described
  • At different times t1, t2, t3, t4, the object passes through different positions; the velocity at each exact time is the instantaneous velocity.
  • Velocity is a vector; it has both magnitude and direction. The direction at any instant is the direction of motion at that moment.
• Relation to average velocity
  • Instantaneous velocity differs from average velocity, which uses displacement over the total time.
  • In calculus, the instantaneous velocity is the slope of the position-time function: $v(t) = \frac{dx}{dt}.$
• Conceptual notes from the lecture
  • The speaker uses arrows to illustrate vinitial, vfinal, and the varying velocity along the path.
  • The instantaneous velocity can be directed to the right (positive) or to the left (negative), depending on the motion.

Speed versus velocity

• Definitions
  • Velocity: a vector quantity with both magnitude and direction.
  • Speed: a scalar quantity that represents the magnitude of velocity (i.e., how fast an object is moving, regardless of direction).
• Instantaneous speed
  • Instantaneous speed is the magnitude of the instantaneous velocity: $\text{speed}_{\text{inst}} = |\vec{v}(t)|.$
• Examples of language usage
  • In everyday language, people say the car’s speed is 50 mph, but in physics, you should specify velocity with a direction (e.g., 50 mph to the east).
• Relationship between speed and velocity in the same instant
  • If the velocity at a moment is positive to the right, its speed is the positive magnitude of that velocity.
  • If the velocity is negative (to the left), the speed is the magnitude of that leftward velocity (a positive number), while the velocity itself remains negative.

Acceleration

• Definition and interpretation
  • Acceleration is the rate of change of velocity with respect to time.
  • It is a vector: $\vec{a} = \frac{\Delta \vec{v}}{\Delta t} = \frac{\vec{v}<em>{\text{final}} - \vec{v}</em>{\text{initial}}}{t<em>{\text{final}} - t</em>{\text{initial}}}.$
• Average acceleration
  • $\vec{a}<em>{\text{avg}} = \frac{\Delta \vec{v}}{\Delta t} = \frac{\vec{v}</em>{\text{final}} - \vec{v}<em>{\text{initial}}}{t</em>{\text{final}} - t_{\text{initial}}}.$
  • Acceleration is a vector; thus both the numerator (change in velocity) and the denominator (time) influence the direction and magnitude.
• Positive vs. negative acceleration
  • If velocity increases in the positive direction, acceleration is positive (to the right, in the given example).
  • If velocity decreases in the positive direction (or increases in the negative direction), acceleration is negative (to the left).
  • The sign of the acceleration indicates whether the velocity is increasing or decreasing along the chosen axis.
• Illustrating with vectors
  • Change in velocity, (\Delta \vec{v} = \vec{v}_{\text{final}} - \vec{\text{initial}}), is a vector; the direction of acceleration follows the direction of this change when divided by time.
• Units
  • Unit of velocity: $\text{m/s}$
  • Unit of time: $\text{s}$
  • Unit of acceleration: $\text{m/s}^2$
• Constant versus varying acceleration
  • In this course, constant acceleration is often assumed for problems: acceleration does not change with time.
  • If acceleration is constant, velocity changes linearly with time: a straight-line change in velocity versus time.
  • Real-world motions can have non-constant acceleration, but the course typically focuses on constant-acceleration problems.
• Special notes from the lecture
  • When velocity increases to the right, acceleration is positive; when velocity decreases (or when the velocity vector points left), acceleration can be negative.
  • The concept of subtracting vectors to obtain (\Delta \vec{v}) is used to determine the direction and magnitude of acceleration.

Key takeaways and relationships

• Summary of quantities (as discussed in the lecture)
  • Displacement ((\Delta \vec{x})) — a vector: final position minus initial position.
  • Distance traveled — a scalar: total length of the path traveled.
  • Average velocity — a vector: displacement divided by total time: $\vec{v}_{\text{avg}} = \frac{\Delta \vec{x}}{\Delta t}.$
  • Instantaneous velocity — a vector: velocity at a specific time; conceptually $\vec{v}(t) = \frac{d\vec{x}}{dt}.$
  • Speed — a scalar: magnitude of velocity; instantaneous speed is the magnitude of the instantaneous velocity: $\text{speed}_{\text{inst}} = |\vec{v}(t)|.$
  • Average speed — a scalar: distance traveled divided by total time.
  • Acceleration — a vector: rate of change of velocity: $\vec{a}_{\text{avg}} = \frac{\Delta \vec{v}}{\Delta t}.$
• Practical implications
  • In physics, velocity and acceleration carry direction; speed and distance are scalar magnitudes.
  • Directional signs are essential for correctly describing motion (positive vs. negative along a chosen axis).
  • When considering a real journey with varying speeds, average velocity uses net displacement over total time, while average speed uses total distance over total time.
  • For problems, it is common to assume constant acceleration to simplify calculations unless otherwise specified.

Quick formulas to remember

• Average velocity
  • $\vec{v}<em>{\text{avg}} = \frac{\Delta \vec{x}}{\Delta t} = \frac{\vec{x}</em>{\text{final}} - \vec{x}<em>{\text{initial}}}{t</em>{\text{final}} - t_{\text{initial}}}$
• Average acceleration
  • $\vec{a}<em>{\text{avg}} = \frac{\Delta \vec{v}}{\Delta t} = \frac{\vec{v}</em>{\text{final}} - \vec{v}<em>{\text{initial}}}{t</em>{\text{final}} - t_{\text{initial}}}$
• Speed vs. velocity
  • $\text{speed}<em>{\text{inst}} = |\vec{v}(t)|, \quad \text{velocity}</em>{\text{inst}} = \vec{v}(t)$
• Units
  • Velocity and speed: $\text{m/s}$
  • Acceleration: $\text{m/s}^2$

End of notes

• The content covered: displacement, distance traveled, average velocity, instantaneous velocity, speed, average speed, acceleration (and the concept of constant acceleration as a common simplifying assumption).
• These concepts will underpin more advanced kinematics equations and problems later in the course.