Physics with Calculus: Motion in One and Two Dimensions & Vectors

Motion in One Dimension

Describing Motion

Motion in one dimension involves tracking the position, velocity, and acceleration of an object along a straight line. The position is specified relative to a coordinate system, and changes in position are called displacement.

• Position (x): The location of an object relative to a reference point.

• Displacement (Δx): The change in position, a signed quantity: $\Delta x = x_f - x_i$

• Time interval (Δt): The elapsed time between two events: $\Delta t = t_f - t_i$

Velocity and Acceleration

Velocity is the rate of change of position, and acceleration is the rate of change of velocity. Both are vector quantities, meaning they have magnitude and direction.

• Velocity (v): $v = \frac{\Delta x}{\Delta t}$

• Acceleration (a): $a = \frac{\Delta v}{\Delta t}$

• Units: Velocity in m/s, acceleration in m/s2

Graphical Representations

Position-versus-time and velocity-versus-time graphs are essential tools for visualizing motion.

• Position vs. Time: The slope of the graph gives velocity.

• Velocity vs. Time: The slope gives acceleration; the area under the curve gives displacement.

Uniform Motion

Uniform motion occurs when velocity is constant (acceleration is zero). The position changes linearly with time.

• Kinematic equation: $x_f = x_i + v \Delta t$

Motion with Constant Acceleration

When acceleration is constant, velocity changes linearly and position changes quadratically with time.

• Kinematic equations:

• $v_f = v_i + a \Delta t$

• $x_f = x_i + v_i \Delta t + \frac{1}{2} a (\Delta t)^2$

• $v_f^2 = v_i^2 + 2a \Delta x$

Sign of Acceleration

The sign of acceleration depends on the direction of motion and whether the object is speeding up or slowing down.

• Speeding up: velocity and acceleration have the same sign.

• Slowing down: velocity and acceleration have opposite signs.

Example: Coming to a Stop in a Car

When a car brakes to a stop, its velocity decreases steadily, and the acceleration is negative (opposite to the direction of motion).

• Initial velocity: $v_i = 15$ m/s

• Final velocity: $v_f = 0$ m/s

• Time interval: $\Delta t = 1.5$ s

• Acceleration: $a = \frac{v_f - v_i}{\Delta t}$

• Displacement: $\Delta x = v_i \Delta t + \frac{1}{2} a (\Delta t)^2$

Quadratic Relationships

When position changes as the square of time, the graph is a parabola. This is typical for constant acceleration motion.

• $y = A x^2$

• Doubling x increases y by a factor of 4.

Free Fall

Free fall is motion under the influence of gravity alone. All objects in free fall have the same acceleration, regardless of mass.

• Free-fall acceleration: $g = 9.8$ m/s2 (downward)

• Use kinematic equations with $a = -g$

Example: Analyzing a Rock’s Fall

A rock dropped from rest falls 100 m. The time to fall and final velocity can be found using kinematic equations for free fall.

• Initial position: $y_i = 100$ m

• Initial velocity: $v_i = 0$ m/s

• Acceleration: $a = -g$

• Use $y_f = y_i + v_i \Delta t + \frac{1}{2} a (\Delta t)^2$ to solve for $\Delta t$

• Use $v_f = v_i + a \Delta t$ to find final velocity

Example: Springbok Leap

A springbok accelerates upward and then enters free fall. The speed at takeoff and the height of the leap are found using constant acceleration equations.

• Phase 1: Upward acceleration

• Phase 2: Free fall

• Use $v_f^2 = v_i^2 + 2a \Delta y$ for both phases

Vectors and Motion in Two Dimensions

Vectors: Definition and Representation

A vector is a quantity with both magnitude and direction. Examples include displacement, velocity, and acceleration.

• Vector notation: $\vec{A}$

• Magnitude: $A$ (always positive)

• Vectors are equal if they have the same magnitude and direction, regardless of their starting point.

Vector Addition and Subtraction

Vectors can be added using the tip-to-tail method or the parallelogram rule. Subtraction is performed by adding the negative of a vector.

• Tip-to-tail: Place the tail of one vector at the tip of another.

• Parallelogram rule: Draw both vectors from a common origin; the diagonal is the sum.

Multiplication by a Scalar

Multiplying a vector by a positive scalar changes its magnitude but not its direction. Multiplying by a negative scalar reverses its direction.

• Zero vector: Multiplying by zero gives a vector of zero length.

Coordinate Systems and Vector Components

Vectors can be decomposed into components along coordinate axes, usually Cartesian (x, y).

• Component vectors: $A_x$ and $A_y$

• Finding components: $A_x = A \cos \theta$, $A_y = A \sin \theta$

• Magnitude from components: $A = \sqrt{A_x^2 + A_y^2}$

• Angle from components: $\theta = \tan^{-1}(A_y / A_x)$

Working with Components

Vector addition and subtraction can be performed by adding or subtracting corresponding components.

• $\vec{A} + \vec{B} = (A_x + B_x) \hat{i} + (A_y + B_y) \hat{j}$

Applications: Projectile and Circular Motion

Projectile motion and circular motion are examples of two-dimensional motion analyzed using vectors and their components.

• Projectile motion: Parabolic trajectory, analyzed with horizontal and vertical components.

• Circular motion: Constant speed but changing direction, resulting in acceleration.

Summary Table: Kinematic Equations for Constant Acceleration

Equation	Physical Meaning
$v_f = v_i + a \Delta t$	Final velocity after time interval
$x_f = x_i + v_i \Delta t + \frac{1}{2} a (\Delta t)^2$	Position after time interval
$v_f^2 = v_i^2 + 2a \Delta x$	Final velocity from displacement

Additional info: These notes expand on brief points from the slides, providing full academic context, definitions, and examples. Images included are directly relevant to the explanations and reinforce key concepts.