Kinematics: Vectors, Scalars, and Motion

Vectors and Scalars

• Definition of Vectors and Scalars:

  • Vectors: These possess both magnitude and direction.

  • Scalars: These possess only magnitude.

• Vector Analysis:

  • It is essential to know basic trigonometric rules and how to decompose vectors into components.

• Examples of Vectors vs. Scalars:

  • Acceleration: Vector

  • Velocity: Vector

  • Momentum: Vector

  • Force: Vector

  • Time: Scalar

  • Volume: Scalar

  • Speed: Scalar

  • Mass: Scalar

• Contextual Examples:

  • A car traveling at $60\,mph$ to the east is described by a vector (velocity).

  • A car traveling at $60\,mph$ is described by a scalar (speed).

Displacement and Distance Traveled

• Displacement: Defined as how "out of place" an object is. Displacement is a vector quantity.

• Distance: Defined as how far something traveled to get "out of place." Distance is a scalar quantity.

Average Speed and Velocity

• Average Speed: The average speed of travel over a time $t$ is calculated using the formula:

  • $\text{Average Speed} = \frac{s}{t}$

  • In this formula, $s$ represents the total distance traveled and $t$ is the duration of travel.

• Average Velocity: The average velocity of travel over a time $t$ is calculated using the formula:

  • $\text{Average Velocity} = \frac{s}{t}$

  • In this formula, $s$ represents the total displacement and $t$ is the duration of travel.

• Practical Note: In problem-solving, it is crucial to read carefully as there is a distinct difference between speed and velocity.

Instantaneous Speed and Velocity

• Instantaneous Speed: This is the speed of an object at a specific instant in time. It is given by the formula:

  • $\text{v} = \frac{\Delta s}{\Delta t}$

  • $\Delta s$ is the distance traveled during an extremely short time period $\Delta t$ surrounding that instant.

• Instantaneous Velocity: This is the velocity of an object at a specific instant in time. It is given by the formula:

  • $\text{v} = \frac{\Delta s}{\Delta t}$

  • $\Delta s$ is the displacement of the object during an extremely short time period $\Delta t$ surrounding that instant.

• Calculus Definitions:

  • Because instantaneous values change as time passes, the interval $\Delta t$ must be infinitesimally short.

  • Instantaneous Speed: Calculated as the derivative of the distance with respect to time ($t$):

    • $\text{Speed} = \frac{ds}{dt}$

  • Instantaneous Velocity: Calculated as the derivative of the displacement with respect to time ($t$):

    • $\text{Velocity} = \frac{ds}{dt}$

Acceleration

• Definition: Acceleration is the changing of an object’s velocity with time. It is a vector quantity.

• Dimensions and Units: Acceleration has dimensions of length per time squared. Its common units are meters per second squared ($m/s^{2}$).

• Average Acceleration ($\bar{a}$): Defined as the change of velocity over the change in time:

  • $\bar{a} = \frac{\Delta v}{\Delta t} = \frac{v_{f} - v_{i}}{t_{f} - t_{i}}$

• Instantaneous Acceleration ($a$): The acceleration of an object at a specific point in time. It is expressed as the first derivative of velocity with respect to time ($t$) or the second derivative of displacement with respect to time ($t$):

  • $a = \frac{dv}{dt} = \frac{d^{2}s}{dt^{2}}$

1D Motion with Constant Velocity

• Graphs for Constant Velocity:

  • The $s$ vs $t$ (displacement vs. time) graph for 1D motion with constant velocity is a straight line.

  • The slope of the graph, by definition, represents the velocity, which is constant in this scenario.

• Calculation of Final Position:

  • $s = s_{0} + vt$

1D Motion with Constant Acceleration

• General Properties:

  • Acceleration ($a$) is a vector. In 1D motion, it is represented by a single real number with a positive or negative sign.

  • When a > 0, the velocity is increasing.

  • When a < 0, the velocity is decreasing.

• Graph for Constant Acceleration:

  • The graph of $v$ vs $t$ (velocity vs. time) will be a straight line when acceleration is constant.

The Big Four Kinematics Equations

These formulas describe the motion of an object with constant acceleration, relating velocity, time, displacement, and acceleration:

• Equation 1: $v_{f} = v_{0} + at$

• Equation 2: $s = v_{0}t + \frac{1}{2}at^{2}$

• Equation 3: $v_{f}^{2} = v_{0}^{2} + 2as$

• Equation 4: $s = \frac{v_{0} + v_{f}}{2} t$

Free Fall and Gravitational Acceleration

• Universal Acceleration: In a vacuum, all objects (e.g., an apple and a bowling ball) experience the same acceleration due to Earth's gravitational pull.

• Standard Value of Gravity ($g$): All objects on Earth experience roughly the same gravitational acceleration, where $g = 9.81\,m/s^{2}$.

• Direction and Signs: The sign of gravitational acceleration (positive or negative) depends on which direction is defined as positive in a given problem.

• Regional Variations:

  • Location affects gravitational acceleration. In Boston, for instance, the value is $9.80\,m/s^{2}$.

  • Acceleration differs on other planets.

• Exam Shortcuts:

  • Using the approximate value $g = 10\,m/s^{2}$ is highly suggested for competitive physics tests like the $F=ma$ exam to facilitate quicker calculations.

  • This shortcut is also acceptable on $AP$ exams, provided the specific value of $g$ used is noted in Free Response Questions ($FRQ$).

Vertical Projectiles

• Scenario: A ball thrown straight up into the air is a vertical projectile.

• Modeling Vertical Motion:

  • When setting the "up" direction as positive, the equations of motion mirror the Big Four kinematics equations.

Terminal Velocity

• Observation: Balloons and feathers drift down slowly at a constant velocity because they reach their terminal velocities quickly.

• Equilibrium State: Terminal velocity is reached when the buoyancy force, air friction, and gravity are at equilibrium.

• Behavior at Terminal Velocity: Once an object reaches terminal velocity, it maintains that constant velocity and can no longer accelerate due to gravity.

• Real-World Example: Skydivers Typically have a terminal velocity of approximately $200\,km/hr$.

Questions & Discussion

• Velocity vs. Time Graphs: What would a $v$ vs $t$ graph look like for motion with constant velocity?

• Area Under the Curve: What does the area under the curve of a $v$ vs $t$ graph signify?

• Projectile Exercise: Derive the equation for the time it takes for a projectile to reach the peak of its trajectory. Additionally, determine the maximum height reached by the projectile.