Kinematics and Newton's Laws: Exam 1 Review Study Notes

Kinematics

Displacement, Velocity, and Acceleration

Kinematics is the study of motion without considering its causes. The primary quantities are displacement, velocity, and acceleration.

• Displacement (\(\Delta x\)): The change in position of an object. It is a vector quantity, having both magnitude and direction. $\Delta x = x_f - x_i$ Where $x_f$ is the final position and $x_i$ is the initial position.

• Velocity (\(v\)): The rate of change of displacement with respect to time. It is also a vector. $v = \frac{\Delta x}{\Delta t} = \frac{x_f - x_i}{\Delta t}$ Note: When motion is uniform (constant velocity), this formula applies directly. For non-uniform motion, use instantaneous velocity.

• Speed: The magnitude of velocity; always positive and does not include direction.

• Acceleration (\(\vec{a}\)): The rate of change of velocity with respect to time. $\vec{a} = \frac{\Delta \vec{v}}{\Delta t} = \frac{\vec{v}_f - \vec{v}_i}{\Delta t}$

Key Points:

• When velocity and acceleration are parallel, speed changes.

• When velocity and acceleration are perpendicular, direction changes.

• On a position vs. time graph, the slope gives velocity.

• On a velocity vs. time graph, the slope gives acceleration.

Example: If a car moves from $x_i = 0$ m to $x_f = 20$ m in 4 s, its average velocity is $v = \frac{20\,\text{m} - 0\,\text{m}}{4\,\text{s}} = 5\,\text{m/s}$.

Kinematic Equations

Equations for Constant Acceleration

When acceleration is constant, the following kinematic equations relate displacement, velocity, acceleration, and time:

• $v_f = v_i + a t$

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

• $v_f^2 = v_i^2 + 2 a \Delta x$

These equations can be applied separately to the x and y components of motion, especially in two-dimensional problems (e.g., projectile motion). Time is typically the same for both directions.

• X-direction:

  • $v_{x,f} = v_{x,i} + a_x t$

  • $\Delta x = v_{x,i} t + \frac{1}{2} a_x t^2$

  • $v_{x,f}^2 = v_{x,i}^2 + 2 a_x \Delta x$

• Y-direction:

  • $v_{y,f} = v_{y,i} + a_y t$

  • $\Delta y = v_{y,i} t + \frac{1}{2} a_y t^2$

  • $v_{y,f}^2 = v_{y,i}^2 + 2 a_y \Delta y$

Example: A ball is thrown upward with $v_{y,i} = 10\,\text{m/s}$ and $a_y = -9.8\,\text{m/s}^2$. After 1 s, $v_{y,f} = 10 - 9.8 \times 1 = 0.2\,\text{m/s}$.

Newton's Laws of Motion

Fundamental Principles of Dynamics

Newton's Laws describe the relationship between forces and motion.

• Newton's Second Law: The net force acting on an object is equal to its mass times its acceleration. $\vec{F}_{\text{net}} = m \vec{a}$

• Vector Addition: To find the net force, add the components of all force vectors.

• Direction: Acceleration and net force are always in the same direction. A net force in the x-direction causes acceleration in the x-direction, and similarly for y.

• Newton's First Law (Inertia): An object will maintain its velocity unless acted upon by a net external force.

• Newton's Third Law: For every action, there is an equal and opposite reaction. These forces act on different objects (e.g., the force of A on B is paired with the force of B on A).

Example: If a 2 kg object experiences a net force of 10 N to the right, its acceleration is $a = \frac{F_{\text{net}}}{m} = \frac{10}{2} = 5\,\text{m/s}^2$ to the right.

Problem Solving Tips

Strategies for Kinematics and Dynamics Problems

• If acceleration is not constant, divide the motion into segments where acceleration is constant and solve each segment separately.

• For projectile motion:

  • $a_x = 0$ (no horizontal acceleration)

  • $a_y = -9.8\,\text{m/s}^2$ (vertical acceleration due to gravity)

  • Initial vertical velocity $v_{y,i}$ may be given or calculated.

• On an inclined plane (just gravity): $a = \pm g \sin \theta$ (use a tilted coordinate system for easier analysis).

• Draw clear diagrams, identify all given information, and resolve vectors into x and y components.

• Practice interpreting and using motion graphs (position vs. time, velocity vs. time, acceleration vs. time).

• Ask clarifying questions if unsure about a problem statement or concept.

Example: For a projectile launched horizontally from a height, use $a_x = 0$ and $a_y = -9.8\,\text{m/s}^2$ to analyze motion in each direction separately.

Additional info: These notes summarize foundational concepts in introductory college physics, focusing on kinematics and Newton's Laws, and provide strategies for solving typical exam problems.