Kinematics and Dynamics: Study Notes for College Physics

Kinematics: Motion in One and Two Dimensions

Uniform Motion

Uniform motion describes the movement of an object at a constant velocity, meaning its speed and direction remain unchanged over time.

• Position Equation: $x_f = x_i + v_x \Delta t$

• Velocity: The rate of change of position; for uniform motion, $v_x$ is constant.

• Example: A car moving at 20 m/s for 5 seconds covers $x_f = x_i + 20 \times 5 = x_i + 100$ meters.

Constant Acceleration

When an object's velocity changes at a constant rate, it is said to be under constant acceleration. This is common in free-fall and projectile motion.

• Position Equation: $x_f = x_i + v_{ix} \Delta t + \frac{1}{2} a_x \Delta t^2$

• Velocity Equation: $v_{fx} = v_{ix} + a_x \Delta t$

• Displacement Equation: $v_{fx}^2 = v_{ix}^2 + 2 a_x \Delta x$

• Example: A ball thrown upward with $v_{ix} = 10$ m/s and $a_x = -9.8$ m/s2 will reach its maximum height when $v_{fx} = 0$.

Variable Acceleration

If acceleration is not constant, calculus is required to determine velocity and position.

• Velocity: $v_f = v_i + \int_{t_i}^{t_f} a_x(t) dt$

• Position: $x_f = x_i + \int_{t_i}^{t_f} v_x(t) dt$

• Example: If $a_x(t) = 4t$, then $v_x(t) = \int 4t dt = 2t^2 + C$.

Free-Fall Motion

Objects in free-fall experience constant acceleration due to gravity, $g = 9.8$ m/s2 downward.

• Vertical Position: $y_f = y_i + v_{iy} \Delta t + \frac{1}{2} g \Delta t^2$

• Vertical Velocity: $v_{fy} = v_{iy} + g \Delta t$

• Example: Dropping a ball from $y_i = 10$ m, $v_{iy} = 0$, after $t = 1.43$ s, $y_f = 0$.

Kinematics in Two Dimensions

Motion in two dimensions involves both x and y components, often analyzed separately.

• Position Equations:

  • $x_f = x_i + v_{ix} \Delta t + \frac{1}{2} a_x \Delta t^2$

  • $y_f = y_i + v_{iy} \Delta t + \frac{1}{2} a_y \Delta t^2$

• Velocity Components:

  • $v_{fx} = v_{ix} + a_x \Delta t$

  • $v_{fy} = v_{iy} + a_y \Delta t$

• Example: A projectile launched at an angle has $a_x = 0$, $a_y = -g$.

Projectile Motion

Projectile motion is a special case of two-dimensional motion where the only acceleration is due to gravity.

• Acceleration: $\overrightarrow{a} = -g \hat{j}$

• Initial Velocity Components:

  • $v_{ix} = v_i \cos \theta$

  • $v_{iy} = v_i \sin \theta$

• Example: A rock thrown upward to hit a drone moving horizontally; calculations involve both x and y equations.

Relative Motion

Relative motion describes how the velocity of an object appears from different reference frames.

• Velocity Addition: $\overrightarrow{V}_{CB} = \overrightarrow{V}_{CA} + \overrightarrow{V}_{AB}$

• Example: If a drone moves at 4 m/s and you throw a rock upward, the relative velocity must be considered to land on the drone.

Dynamics: Force and Motion

Newton's Second Law in Two Dimensions

Newton's Second Law relates the net force acting on an object to its acceleration.

• Vector Form: $\overrightarrow{F}_{net} = m \overrightarrow{a}$

• Component Form:

  • $F_{net,x} = m a_x$

  • $F_{net,y} = m a_y$

• Example: A block sliding up an incline; forces are resolved into x and y components.

Friction and Drag Forces

Friction opposes motion between surfaces, while drag opposes motion through a fluid.

• Kinetic Friction: $F_{k} = \mu_k F_n$

• Static Friction: $F_{s} \leq \mu_s F_n$

• Drag Force: $F_{drag} = \frac{1}{2} c \rho A v^2$

• Example: A block sliding up a 30° slope with $\mu_k = 0.1$ and $v_i = 10$ m/s; calculate how far it goes before stopping.

Inclined Plane Problems

Objects on inclined planes experience forces due to gravity, friction, and normal force.

• Gravity Components:

  • Parallel: $F_{g,\parallel} = m g \sin \theta$

  • Perpendicular: $F_{g,\perp} = m g \cos \theta$

• Normal Force: $F_n = m g \cos \theta$

• Friction Force: $F_{k} = \mu_k F_n$

• Example: For $\theta = 30^\ ext{o}$, $\mu_k = 0.1$, $v_i = 10$ m/s, find distance before stopping using $v_{fx}^2 = v_{ix}^2 + 2 a_x \Delta x$.

Sample Calculations and Applications

Several example problems illustrate the application of kinematics and dynamics principles.

• Projectile to Hit a Moving Drone:

  • Drone at 10 m height, 20 m away, moving at 4 m/s.

  • Rock thrown upward; solve for initial speed using kinematic equations.

• Dropping a Ball from a Plane:

  • Plane at 100 m height, speed 30 m/s.

  • Find where to drop the ball to hit a target using $\Delta x = v_{ix} \Delta t$.

• Particle with Time-Dependent Velocity:

  • $v_x(t) = 2t^2 - 4t$

  • Find acceleration: $a_x(t) = \frac{d}{dt} v_x(t) = 4t - 4$

  • Find distance covered after 2 s: $\Delta x = \int_0^2 (2t^2 - 4t) dt = \frac{2}{3} t^3 - 2t^2 \Big|_0^2 = \frac{16}{3} - 8 = -\frac{8}{3}$ m (Additional info: negative sign indicates direction)

Summary Table: Forces on an Inclined Plane

This table summarizes the main forces acting on a block on an inclined plane.

Force	Equation	Direction
Gravity (parallel)	$F_{g,\parallel} = m g \sin \theta$	Down the slope
Gravity (perpendicular)	$F_{g,\perp} = m g \cos \theta$	Into the slope
Normal Force	$F_n = m g \cos \theta$	Perpendicular to surface
Kinetic Friction	$F_k = \mu_k F_n$	Opposes motion

Additional info: Some equations and steps were inferred from context and standard physics principles to ensure completeness and clarity.