Motion in 1D and Vectors: Study Notes

Motion in 1D and Vectors

Velocity

Velocity is a fundamental concept in kinematics, describing the rate of change of an object's position with respect to time. It is a vector quantity, meaning it has both magnitude and direction.

• Average Velocity: Defined as the total displacement divided by the total time taken. For a trip with varying speeds, average velocity is calculated as $v_{avg} = \frac{\Delta x}{\Delta t}$.

• Instantaneous Velocity: The velocity of an object at a specific moment in time, given by the derivative of position with respect to time: $v = \frac{dx}{dt}$.

• Example: If a person drives 10 km east in 0.5 hours, the average velocity is $v_{avg} = \frac{10\,\text{km}}{0.5\,\text{h}} = 20\,\text{km/h}$ east.

Motion Under Constant Acceleration

When an object moves with constant acceleration, its velocity changes at a uniform rate. The equations of motion for constant acceleration are essential for solving kinematics problems.

• Key Equations:

  • $v = v_0 + at$

  • $x = x_0 + v_0 t + \frac{1}{2} a t^2$

  • $v^2 = v_0^2 + 2a(x - x_0)$

• Example: A car accelerates from rest at $2.0\,\text{m/s}^2$ for $5.0\,\text{s}$. Its final velocity is $v = 0 + (2.0)(5.0) = 10.0\,\text{m/s}$.

Free Fall

Free fall refers to the motion of objects under the influence of gravity alone, typically near Earth's surface. The acceleration due to gravity is denoted by $g$ and is approximately $9.8\,\text{m/s}^2$ downward.

• Equations for Free Fall:

  • $y = y_0 + v_0 t - \frac{1}{2} g t^2$

  • $v = v_0 - g t$

• Example: If a ball is dropped from rest ($v_0 = 0$) from a height of $20\,\text{m}$, the time to reach the ground is found by solving $0 = 20 - \frac{1}{2}(9.8)t^2$.

Vectors

Vectors are quantities that have both magnitude and direction. They are essential in physics for representing displacement, velocity, force, and other directional quantities.

• Vector Addition: Vectors can be added graphically (tip-to-tail method) or algebraically using components.

• Components of a Vector: Any vector $\vec{A}$ can be broken into $x$ and $y$ components: $A_x = A \cos \theta$, $A_y = A \sin \theta$.

• Resultant Vector: The sum of two or more vectors. For vectors $\vec{A}$ and $\vec{B}$, $\vec{R} = \vec{A} + \vec{B}$.

• Example: If $|\vec{F}_1| = 80\,\text{N}$ at $60^\circ$ above the $x$-axis and $|\vec{F}_2| = 120\,\text{N}$ along the $x$-axis, their components can be calculated and summed to find the resultant force.

Vector Diagram Example

The diagram shows two forces acting at angles, illustrating how to resolve vectors into components and find the resultant.

• Force Components:

  • $F_{1x} = 80 \cos 60^\circ = 40\,\text{N}$

  • $F_{1y} = 80 \sin 60^\circ = 69.3\,\text{N}$

  • $F_{2x} = 120\,\text{N}$ (along $x$-axis)

  • $F_{2y} = 0$

• Resultant Force: $F_{Rx} = F_{1x} + F_{2x}$, $F_{Ry} = F_{1y} + F_{2y}$

• Magnitude: $|\vec{F}_R| = \sqrt{F_{Rx}^2 + F_{Ry}^2}$

Summary Table: Vector Components

Vector	Magnitude (N)	Angle	$x$-Component (N)	$y$-Component (N)
$\vec{F}_1$	80	60°	40	69.3
$\vec{F}_2$	120	0°	120	0

Additional info: These notes expand on the brief questions by providing definitions, equations, and worked examples relevant to introductory college physics topics in kinematics and vectors.