one dimensional motion

One-Dimensional Motion (Rectilinear)

One-dimensional (rectilinear) motion is motion along a straight line; the object can move forward or backward along that line (or up and down in a vertical sense).

Two- and Three-Dimensional Motion

Two-dimensional motion occurs in a plane, e.g., moving straight down a road and turning left, or a ball curving in flight. Three-dimensional motion involves changes in direction in space, described by coordinates
(x, y, z) .

Frame of Reference, Position, Displacement and Distance

When an object moves, its position changes relative to a chosen reference point. A reference point (origin) in a frame of reference is needed to describe position. A frame of reference is a coordinate system used to measure position and other properties.

Frame of Reference

A frame of reference is a coordinate system used to represent and measure properties of objects, such as position.

Position in a Frame of Reference

Position is the place of an object relative to the reference point. It can be positive or negative and is a vector quantity.

Distance vs Displacement

Distance is the total length traveled and is a scalar. Displacement is the straight-line change in position from initial to final position and is a vector. Displacement is independent of the path taken and points from the initial to the final position.

Example: Displacement

If Samuel walks 80 m east to a shop, his distance is 80 m and his displacement is 80 m east. If he then rests 30 m from home and later returns toward home, his total distance may be greater than the net displacement, which is the straight-line vector from start to finish.

Speed and Velocity

If speed is constant, equal distances are covered in equal times; speed is a scalar. Average speed is total distance over the time interval:
v_{ ext{avg}} = rac{D}{t}
Distance is also related to speed by
D = v \, t
Time is
t = rac{D}{v}

Velocity includes both magnitude and direction; its direction is the same as the displacement. Velocity is a vector:
\vec{v} = \frac{\Delta \vec{x}}{\Delta t}

Conversions

Conversions between distance units:
1\ \text{km} = 1000\ \text{m}
1\ \text{m} = 100\ \text{cm}
1\ \text{m} = 1000\ \text{mm}
To convert time units:
1\ \text{hour} = 3600\ \text{s}

Speed–Velocity Conversions

1\ \text{m s}^{-1} = 3.6\ \text{km h}^{-1}

James Example: Average Speed vs. Average Velocity

If a person walks 2 km away and then 2 km back along the same path in 30 min each (total distance 4 km, total time 60 min), then

• Average speed:
  v_{ ext{avg}} = \frac{4\ \text{km}}{1\ \text{h}} = 4\ \text{km h}^{-1}

• Average velocity: the net displacement is zero (return to start), so
  \vec{v}_{\text{avg}} = 0

Instantaneous Speed vs Instantaneous Velocity

Instantaneous speed is the magnitude of instantaneous velocity; that is,
|\,\vec{v}(t)\,| = \text{instantaneous speed}

Acceleration

Acceleration is the rate of change of velocity; it is a vector and is measured in \text{m s}^{-2}. If the velocity changes, the object accelerates:
\vec{a} = \frac{\Delta \vec{v}}{\Delta t}

Signs of Acceleration

Acceleration can be positive or negative. A negative value does not necessarily mean the object is moving backward; it indicates deceleration in the positive direction or acceleration in the negative direction, depending on the chosen reference.

Examples of Acceleration

A taxi traveling at 20\ \text{m s}^{-1} stopping in 15\ \text{s} has
a = \frac{0 - 20}{15} = -\tfrac{4}{3} \text{ m s}^{-2}
An aircraft accelerating from 100\ \text{m s}^{-1} to 400\ \text{m s}^{-1} in 100\ \text{s} has
a = \frac{400 - 100}{100} = 3\ \text{m s}^{-2}

Equations of Motion (Constant Acceleration)

Under constant acceleration in a straight line, the following apply:

vf = vi + a t \
s = vi t + \tfrac{1}{2} a t^2 \ vf^2 = vi^2 + 2 a s Where: $vi$ is initial velocity, $v_f$ is final velocity, $a$ is acceleration, and $s$ is displacement.

Conditions for Using the Equations

These equations are valid only for motion with constant acceleration in a straight line; they do not apply to circular motion, wave motion, or pendulums.

Problem-Solving Steps

1) List all given quantities and the quantity to be found. 2) Treat directions; take the initial direction as positive. 3) If the object starts from rest, set $vi = 0$. 4) If the object comes to rest, set $vf = 0$ when appropriate. 5) If velocity increases, acceleration is in the direction of motion (positive). 6) Write the equation and substitute values (without units). 7) Provide the final answer with units and direction.

Example Problem Structure

For a motorcycle starting from rest with $a = 3\ \text{m s}^{-2}$, after $t = 6\ \text{s}$, the velocity is
vf = vi + a t = 0 + 3 \times 6 = 18\ \text{m s}^{-1}
The distance traveled in 6 s is
s = v_i t + \tfrac{1}{2} a t^2 = 0 \times 6 + \tfrac{1}{2} \times 3 \times 6^2 = 54\ \text{m}
The average velocity over the first 6 s is
\bar{v} = \frac{s}{t} = \frac{54}{6} = 9\ \text{m s}^{-1}

Quick Reference Equations

• Average speed: v_{ ext{avg}} = \frac{D}{t}

• Distance–Time: D = v t

• Time–Distance: t = \frac{D}{v}

• Instantaneous velocity: \vec{v} = \frac{\Delta \vec{x}}{\Delta t}

• Acceleration: \vec{a} = \frac{\Delta \vec{v}}{\Delta t}

• Constant-acceleration kinematics:
  vf = vi + a t, \quad s = vi t + \tfrac{1}{2} a t^2, \quad vf^2 = v_i^2 + 2 a s