Section 1.1: 1D Motion
In this section, we introduce motion in one dimension. We analyze displacement, velocity, and acceleration, and their relationships through time. Understanding these concepts lays the foundation for all mechanics problems.
Displacement:
Displacement is the change in position: \( \Delta x = x_f - x_i \)
Velocity:
Average velocity: \( v_{avg} = \frac{\Delta x}{\Delta t} \)
Instantaneous velocity: \( v = \frac{dx}{dt} \)
Acceleration:
Average acceleration: \( a_{avg} = \frac{\Delta v}{\Delta t} \)
Instantaneous acceleration: \( a = \frac{dv}{dt} \)
Example 1
A car moves along a straight road. Its position changes from \(x_i = 0 \text{ m}\) to \(x_f = 100 \text{ m}\) in 10 s. Find its average velocity.
\( v_{avg} = \frac{x_f - x_i}{t} = \frac{100 - 0}{10} = 10 \text{ m/s} \)
Practice Problems
- A car travels 50 m in 5 s. Find its average velocity.
- An object moves from 10 m to 30 m in 4 s. Compute the average velocity.
- An object starts at rest and accelerates uniformly at 2 m/s² for 5 s. Find final velocity.
- A runner moves 100 m in 12 s. Determine average speed.
- An object accelerates from 0 to 20 m/s in 10 s. Find average acceleration.