College Physics PHYS: Mechanics - Kinematics Position Displacement Velocity Acceleration Equations of Motion Constant Acceleration Free Fall Projectile Motion

1. What This Is & Why It Matters

Kinematics is the study of the motion of objects without considering the forces that cause the motion. It's the foundation of classical mechanics, and mastering kinematics is essential for understanding more advanced topics like dynamics, energy, and momentum. Without a solid grasp of kinematics, you'll struggle to analyze and solve problems in fields like engineering, physics, and even computer science.

For example, GPS satellites rely on accurate kinematic calculations to correct for time dilation and ensure precise location and time measurements. If you can't understand how to calculate position, velocity, and acceleration, you'll be lost when trying to analyze the motion of these satellites.

2. Key Formulas & Constants

• Position (x): x = x? + v?t + (1/2)at², where x? is the initial position, v? is the initial velocity, t is time, and a is acceleration.
  • Definition: x is the distance from a reference point (usually the origin).
  • Use: Calculate the position of an object at a given time, given its initial position, velocity, and acceleration.
  • Constants: None.
• Displacement (?x): ?x = x - x?, where x is the final position and x? is the initial position.
  • Definition: ?x is the change in position.
  • Use: Calculate the displacement of an object between two points.
  • Constants: None.
• Velocity (v): v = ?x / ?t, where ?x is displacement and ?t is time.
  • Definition: v is the rate of change of position.
  • Use: Calculate the velocity of an object given its displacement and time.
  • Constants: None.
• Acceleration (a): a = ?v / ?t, where ?v is the change in velocity and ?t is time.
  • Definition: a is the rate of change of velocity.
  • Use: Calculate the acceleration of an object given its change in velocity and time.
  • Constants: None.
• Equation of Motion (Constant Acceleration): v² = v?² + 2a(x - x?), where v is final velocity, v? is initial velocity, a is acceleration, x is final position, and x? is initial position.
  • Definition: This equation relates the initial and final velocities, positions, and acceleration of an object under constant acceleration.
  • Use: Calculate the final velocity or position of an object given its initial velocity, position, acceleration, and time.
  • Constants: None.
• Free Fall: v = gt, where v is velocity, g is acceleration due to gravity (approximately 9.8 m/s² on Earth), and t is time.
  • Definition: This equation describes the velocity of an object under the sole influence of gravity.
  • Use: Calculate the velocity of an object in free fall given its time of fall.
  • Constants: g (acceleration due to gravity).
• Projectile Motion: y = x tan(?) - (g/2v?²)(x²), where y is height, x is horizontal distance,-is launch angle, g is acceleration due to gravity, and v? is initial velocity.
  • Definition: This equation describes the height of a projectile as a function of horizontal distance.
  • Use: Calculate the height of a projectile given its horizontal distance, launch angle, and initial velocity.
  • Constants: g (acceleration due to gravity).

3. Step-by-Step Problem-Solving Strategy

1. Draw a free-body diagram: Sketch the object and all the forces acting on it. This will help you identify the forces and choose the correct coordinate system.
2. Choose a coordinate system: Select a coordinate system that aligns with the motion of the object. This will make it easier to apply Newton's laws and solve the problem.
3. Apply Newton's second law: Use the equation F = ma to relate the net force acting on the object to its mass and acceleration.
4. Use kinematic equations: Apply the kinematic equations (position, velocity, acceleration) to solve for the desired quantity.
5. Check your units: Verify that your answer has the correct units.

Common mistakes to avoid:

• Failing to draw a free-body diagram
• Choosing an incorrect coordinate system
• Forgetting to apply Newton's second law
• Using the wrong kinematic equation

4. Common Mistakes & Misconceptions

• Mistake: Assuming that acceleration is always constant.
  • Explanation: Acceleration can change over time due to changing forces or velocities.
  • Right way: Use the equation of motion for constant acceleration, or use the kinematic equations to solve for acceleration.
• Mistake: Failing to account for air resistance.
  • Explanation: Air resistance can significantly affect the motion of objects, especially at high speeds.
  • Right way: Include air resistance in your free-body diagram and apply the correct forces to the object.
• Mistake: Using the wrong kinematic equation.
  • Explanation: Make sure to choose the correct equation based on the given information and the desired quantity.
  • Right way: Review the kinematic equations and choose the correct one for the problem at hand.

5. Exam / Test-Taking Tips

• Multiple choice: Pay attention to the units and limits of the answer choices.
• Free response: Make sure to show all your work and explain your reasoning.
• Conceptual vs. plug-and-chug: Focus on understanding the underlying concepts and principles, rather than just memorizing formulas.
• Trap distinctions: Be aware of common traps like velocity vs. speed, power vs. energy, and resistance vs. resistivity.

6. Quick Practice Problems

Problem 1: Position and Velocity

A car starts from rest and accelerates uniformly to a final velocity of 25 m/s in 5 seconds. Find its position and velocity after 3 seconds.

Solution:

1. Draw a free-body diagram: Not applicable, as there are no forces acting on the car.
2. Choose a coordinate system: Choose a coordinate system with the origin at the starting point of the car.
3. Apply Newton's second law: Not applicable, as there are no forces acting on the car.
4. Use kinematic equations: Use the equation v = v? + at to find the velocity after 3 seconds.
   • v = 0 + (25/5)(3) = 15 m/s
5. Use kinematic equations: Use the equation x = x? + v?t + (1/2)at² to find the position after 3 seconds.
   • x = 0 + 0(3) + (1/2)(5)(3)² = 22.5 m

Physical reasoning: The car accelerates uniformly from rest to a final velocity of 25 m/s in 5 seconds. After 3 seconds, its velocity is 15 m/s, and its position is 22.5 m from the starting point.

Problem 2: Free Fall

A ball is dropped from a height of 10 m above the ground. Find its velocity after 2 seconds.

Solution:

1. Draw a free-body diagram: Not applicable, as there are no forces acting on the ball.
2. Choose a coordinate system: Choose a coordinate system with the origin at the ground level.
3. Apply Newton's second law: Not applicable, as there are no forces acting on the ball.
4. Use kinematic equations: Use the equation v = gt to find the velocity after 2 seconds.
   • v = 9.8(2) = 19.6 m/s
5. Check your units: Verify that the answer has the correct units (m/s).

Physical reasoning: The ball is under the sole influence of gravity, which accelerates it downward at 9.8 m/s². After 2 seconds, its velocity is 19.6 m/s.

7. Last-Minute Cram Sheet

• Kinematic equations: x = x? + v?t + (1/2)at², v = v? + at, a = ?v / ?t
• Free fall: v = gt
• Projectile motion: y = x tan(?) - (g/2v?²)(x²)
• Acceleration due to gravity: g-9.8 m/s² on Earth
• Acceleration is zero at the top of a projectile's path, but velocity is not!
• Falling objects accelerate downward at 9.8 m/s², regardless of their mass!

8. Further Study Resources

• Textbooks: University Physics by Young & Freedman, Physics for Scientists and Engineers by Serway & Jewett
• Websites: Flipping Physics, Khan Academy, HyperPhysics
• Interactive simulations: PhET, PhysLab