One-Dimensional Kinematics

Introduction

One-dimensional kinematics is the study of motion along a straight line. This topic forms the foundation for understanding more complex motion in physics. The key concepts include position, distance, displacement, speed, velocity, and acceleration, as well as the equations that describe motion under constant acceleration.

Units of Chapter 2

• Position, Distance, and Displacement

• Average Speed and Velocity

• Instantaneous Velocity

• Acceleration

• Motion with Constant Acceleration

• Applications of the Equations of Motion

• Freely Falling Objects

Position, Distance, and Displacement

Defining Motion in One Dimension

• Position is the location of an object along a straight line, specified relative to an origin and a chosen positive direction.

• Distance is the total length of the path traveled, regardless of direction.

• Displacement is the change in position of an object, defined as the final position minus the initial position. Displacement is a vector and can be positive or negative depending on direction.

Example: If you walk from your house to the grocery store (4.3 mi) and then to your friend's house (2.1 mi), your total distance is 6.4 mi, but your displacement is the straight-line distance from your house to your friend's house.

Average Speed and Velocity

Describing How Fast and in What Direction

• Average Speed is the total distance traveled divided by the total time taken:

• Average Velocity is the displacement divided by the elapsed time:

• If you return to your starting point, your displacement is zero, so your average velocity is zero, even if you traveled a nonzero distance.

Example: Driving 8.6 miles in a round trip in 1 hour gives an average speed of 8.6 mi/h, but average velocity is 0 mi/h.

Instantaneous Velocity

Velocity at a Specific Instant

• Instantaneous Velocity is the velocity of an object at a particular moment in time. It is the limit of the average velocity as the time interval approaches zero:

• Graphically, instantaneous velocity is the slope of the tangent to the position vs. time graph at a given point.

Acceleration

Describing Changes in Velocity

• Average Acceleration is the change in velocity divided by the time interval over which the change occurs:

• Instantaneous Acceleration is the acceleration at a specific instant, given by the slope of the velocity vs. time graph at that point.

• Acceleration can be positive (speeding up) or negative (slowing down, also called deceleration), depending on the direction of velocity and acceleration vectors.

Typical Accelerations (m/s2)

Motion with Constant Acceleration

Equations of Motion

• When acceleration is constant, the following equations describe the motion:

• These equations relate position, velocity, acceleration, and time for objects moving in a straight line with constant acceleration.

Example: A car accelerating from rest at 2 m/s2 for 5 seconds will have a final velocity m/s and a displacement m.

Applications of the Equations of Motion

Solving Real-World Problems

• These equations are used to solve problems such as stopping distances, time to reach a certain speed, or distance traveled under constant acceleration.

• Always identify known and unknown variables, choose the appropriate equation, and solve algebraically before substituting numbers.

Freely Falling Objects

Motion Under Gravity

• Free fall refers to motion under the influence of gravity alone, with negligible air resistance.

• The acceleration due to gravity, , is approximately m/s2 downward near Earth's surface, but varies slightly with location.

• Objects in free fall experience constant acceleration, regardless of their mass (neglecting air resistance).

• Equations of motion for free fall are the same as for constant acceleration, with (downward).

Example: Dropping a ball from rest, its velocity after seconds is , and its displacement is .

Summary of Key Concepts

• Distance: Total length of travel (scalar)

• Displacement: Change in position (vector)

• Average speed: Distance divided by time

• Average velocity: Displacement divided by time

• Instantaneous velocity: Slope of position vs. time graph at a point

• Average acceleration: Change in velocity divided by time

• Instantaneous acceleration: Slope of velocity vs. time graph at a point

• Constant acceleration: Use kinematic equations to relate position, velocity, acceleration, and time

• Freely falling objects: Constant downward acceleration m/s2