Physics Fundamentals: Vectors, Kinematics, and Applications – Study Notes

Vectors and Vector Operations

Vector Magnitude and Direction (Cross Product)

Vectors are quantities that have both magnitude and direction. The cross product of two vectors results in a third vector perpendicular to the plane containing the original vectors.

• Definition: The cross product M × N of vectors M and N is given by $|M||N|\sin\theta$, where $\theta$ is the angle between them.

• Direction: Determined by the right-hand rule; in this case, the direction is along the z-axis (+z or -z).

• Example: For M = 3.2 m at 48° from y-axis and N = 2.3 m at 48° from -x-axis, the magnitude is $3.2 \times 2.3 \times \sin(96°) \approx 7.32$ (rounded as per significant figures).

Additional info: The cross product is used in physics to find torque, angular momentum, and magnetic force.

Vector Addition and Subtraction

Vectors can be added or subtracted graphically or algebraically. Subtracting B from A is equivalent to adding A and -B.

• Graphical Method: Place the tail of -B at the head of A; the resultant A - B is drawn from the tail of A to the head of -B.

• Algebraic Method: Subtract corresponding components: $A - B = (A_x - B_x, A_y - B_y)$.

• Example: If A and B are given, A - B is constructed by reversing B and adding to A.

Significant Figures in Calculations

Rules for Significant Figures

Significant figures reflect the precision of measured quantities. When performing calculations, the result should not be more precise than the least precise measurement.

• Division Rule: The result should have the same number of significant figures as the measurement with the fewest significant figures.

• Example: $21.8 / 16.9 = 1.29$ (rounded to three significant figures).

Kinematics: Motion in One and Two Dimensions

Displacement and Velocity in Component Form

Displacement is a vector quantity representing the change in position. When velocity changes direction, displacement must be calculated for each segment and expressed in component form.

• Component Form: $\vec{d} = v_x t \hat{i} + v_y t \hat{j}$

• Example: For a particle with velocity $(17\hat{i} - 6\hat{j})$ mph for 45 min, and then $(5\hat{i} + 18\hat{j})$ mph for 1.5 h, calculate each displacement and sum components.

Estimating Mass Using Density and Volume

Mass can be estimated by treating objects as simple geometric shapes and using density.

• Formula: $\text{Mass} = \text{Density} \times \text{Volume}$

• Volume of Cone: $V = \frac{1}{3} \pi r^2 h$

• Example: For a skyscraper modeled as a cone (height = 900 m, base diameter = 230 m, density = 19 kg/m³), calculate mass.

Average Velocity

Average velocity is the total displacement divided by the total time taken.

• Formula: $v_{avg} = \frac{\Delta x}{\Delta t}$

• Example: If a dog runs 8.2 km to a park and returns in 25 min, average velocity for the return is negative if direction is reversed.

Acceleration from Velocity-Time Graphs

Acceleration is the rate of change of velocity. It can be determined from the slope of a velocity-time graph.

• Formula: $a = \frac{\Delta v}{\Delta t}$

• Example: For a projectile, use the graph to estimate acceleration at specific times by calculating the slope.

Free Fall and Projectile Motion

Objects in free fall experience constant acceleration due to gravity. Projectile motion involves both horizontal and vertical components.

• Vertical Motion: $y = y_0 + v_{y0} t - \frac{1}{2} g t^2$

• Horizontal Motion: $x = x_0 + v_{x0} t$

• Example: A bag dropped from a balloon at 50 m height with initial upward speed; calculate position and speed after 0.5 s.

Relative Motion

Relative velocity is the velocity of one object as observed from another moving object.

• Formula: $v_{A/B} = v_A - v_B$

• Example: Two unicyclists moving in opposite directions; calculate relative speed.

Applications of Kinematics and Dynamics

Overtaking Problems (Uniform Acceleration and Constant Speed)

When one object starts from rest with acceleration and another moves at constant speed, the overtaking point can be found by equating their positions.

• Formula: $x_{accel} = \frac{1}{2} a t^2$, $x_{const} = v t$

• Example: Van Swift starts with $a = 3.2$ m/s², Thunder moves at $v = 22.0$ m/s; find Swift's speed when overtaken.

Deceleration and Acceleration in Terms of g

Acceleration can be expressed in m/s² or as a fraction of gravitational acceleration $g$ ($g = 9.8$ m/s²).

• Formula: $a = \frac{\Delta v}{\Delta t}$, $a_{(g)} = \frac{a}{g}$

• Example: School bus decelerates from constant velocity to rest; calculate acceleration in both units.

Interpreting Graphs: Velocity and Position

Graphs are essential tools for visualizing motion. The highest point on a velocity-time graph indicates maximum velocity; the slope of a position-time graph gives instantaneous velocity.

• Velocity-Time Graph: Maximum velocity at the peak.

• Position-Time Graph: Instantaneous velocity is the slope at a given time.

Projectile Motion: Seed Dispersal and Rock Slides

Projectile motion problems involve decomposing initial velocity into horizontal and vertical components and analyzing motion under gravity.

• Horizontal Component: $v_x = v_0 \cos\theta$

• Vertical Component: $v_y = v_0 \sin\theta$

• Acceleration: $a_x = 0$, $a_y = -g$

• Example: Seed fired at 55 m/s at 54°; find $v_x$, $v_y$, $a_x$, $a_y$ at highest point.

Horizontal Range of Projectiles

The horizontal distance traveled by a projectile is determined by its initial velocity, launch angle, and height above ground.

• Formula: $R = v_x t_{flight}$

• Example: Rock slides off a mountain at 22° below horizontal, initial speed 6.0 m/s, height 13 m; calculate range.

Angle of Projectile After Displacement

The angle of a projectile after traveling a certain horizontal distance can be found using kinematic equations.

• Formula: $\tan\theta = \frac{v_y}{v_x}$ at the given position.

• Example: Football kicked at 30 m/s, 35°; find angle after 75 m horizontal travel.

Velocity and Acceleration from Parametric Equations

Velocity and acceleration can be found by differentiating position functions with respect to time.

• Velocity: $\vec{v}(t) = \frac{d\vec{r}}{dt}$

• Acceleration: $\vec{a}(t) = \frac{d\vec{v}}{dt}$

• Example: Drone velocity given by $\vec{v}(t)$; find magnitude and direction at $t = 3$ s.

Projectile Angle for Targeted Throws

To hit a target at a specific distance and height, the launch angle must be calculated using projectile motion equations.

• Formula: $y = x \tan\theta - \frac{g x^2}{2 v_0^2 \cos^2\theta}$

• Example: Thief throws pouch to partner over a gate; calculate required angle.

Relative Velocity in Two Dimensions

Relative velocity in two dimensions is found by vector subtraction of velocities.

• Formula: $\vec{v}_{blue/red} = \vec{v}_{blue} - \vec{v}_{red}$

• Example: Circus artists on unicycles; find how fast blue appears to move relative to red.