One-Dimensional Kinematics

Introduction to Motion with Constant Acceleration

One-dimensional kinematics studies the motion of objects along a straight line. When acceleration is constant, the equations of motion become especially useful for predicting future positions and velocities. These equations are foundational for understanding more complex motion in physics.

Key Definitions

• Displacement (x): The change in position of an object.

• Velocity (v): The rate of change of displacement with respect to time.

• Acceleration (a): The rate of change of velocity with respect to time.

• Initial velocity (v_0): The velocity at the start of the time interval (t = 0).

• Final velocity (v): The velocity at the end of the time interval (t).

Constant Acceleration Equations of Motion

For motion with constant acceleration, the following equations relate displacement, velocity, acceleration, and time. These equations are derived from the definitions of velocity and acceleration and are essential tools for solving kinematics problems.

Graphical Interpretation of Motion

Graphs are powerful tools for visualizing motion. The area under a velocity versus time graph represents the displacement of an object. For constant acceleration, the velocity-time graph is a straight line, and the area under the curve (a combination of rectangles and triangles) gives the total displacement.

• Area under v-t curve = Displacement

• For constant acceleration, the v-t graph is linear.

Worked Examples

Example 1: Car Accelerating East

A car travels east at 23.0 m/s. Find its velocity after 6.00 s if the acceleration is (a) 1.50 m/s2 east, (b) 2.5 m/s2 west.

• Given: ,

• (a) :

• (b) :

Example 2: Boat Accelerating from Rest

A boat moves at 1.50 m/s, then accelerates at 2.40 m/s2 for 5.00 s. Find (a) its final velocity, (b) the distance traveled.

• (a)

• (b)

Example 3: Area Under Velocity-Time Curve

The distance traveled by the boat in Example 2 can also be found by calculating the area under the velocity-time graph, which consists of a rectangle (initial velocity) and a triangle (change in velocity).

• Area of rectangle:

• Area of triangle:

• Total displacement:

Example 4: Drag Racer's Position Over Time

A drag racer starts from rest and accelerates at 7.40 m/s2. How far has it traveled after 1.00 s, 2.00 s, and 3.00 s?

• Use with , .

• (a) :

• (b) :

• (c) :

Example 5: Police Pursuit Problem

A speeder travels at 17.9 m/s. A police car starts from rest and accelerates at 4.51 m/s2. How long does it take for the police car to catch the speeder, how far have they traveled, and what is the police car's velocity at that moment?

• Set positions equal:

• Speeder:

• Police:

• Equate:

• Distance:

• Police velocity:

Summary Table: Constant-Acceleration Equations

The following table summarizes the five key equations for one-dimensional motion with constant acceleration:

Additional info:

• These equations assume constant acceleration and straight-line motion.

• Choosing a coordinate system and sign convention is crucial for correctly applying the equations.

• Graphical analysis (such as area under velocity-time curves) provides a visual understanding of displacement and velocity changes.