Motion in a Straight Line

The One Thing

Motion in a straight line is analysed by tracking how an object’s position changes with time. Displacement, velocity, and acceleration provide increasingly detailed descriptions of this change, while equations and graphs allow the motion to be calculated and interpreted.

Reactions, Processes and Experiments

In the constant-acceleration equations, $u$ is initial velocity, $v$ is final velocity, $a$ is constant acceleration, $t$ is time, and $x$ is displacement.

SI units are: position, distance, and displacement in metre (m); time in second (s); speed and velocity in metre per second (m/s); and acceleration in metre per second squared (m/s²).

Key Terms

• Reference point: A fixed point or origin used to describe the position and motion of an object.
• Position: The location of an object relative to a chosen origin, represented by a coordinate such as $x$.
• Distance: The total length of the actual path travelled by an object; it is a scalar quantity and is always non-negative.
• Displacement: The change in position of an object, equal to final position minus initial position; it is a vector quantity.
• Speed: The distance travelled per unit time; average speed is total distance divided by total time.
• Velocity: The rate of change of displacement with time; average velocity is total displacement divided by total time.
• Instantaneous velocity: The velocity of an object at a particular instant, given by the rate of change of position with time.
• Acceleration: The rate of change of velocity with time; it may be positive, negative, or zero.
• Uniform motion: Motion in which an object covers equal displacements in equal intervals of time and has constant velocity.
• Non-uniform motion: Motion in which velocity changes with time, either in magnitude, direction, or both.
• Uniform acceleration: Motion in which velocity changes by equal amounts in equal intervals of time.
• Scalar quantity: A physical quantity described only by magnitude, such as distance and speed.
• Vector quantity: A physical quantity described by both magnitude and direction, such as displacement, velocity, and acceleration.
• Position-time graph: A graph showing how position changes with time; its slope gives velocity.
• Velocity-time graph: A graph showing how velocity changes with time; its slope gives acceleration and its area gives displacement.
• Acceleration-time graph: A graph showing how acceleration changes with time; the area under the graph gives the change in velocity.
• Free fall: Motion under the influence of gravity alone, with acceleration approximately equal to $9.8\text{m/s²}$ downward near Earth’s surface.

Easily Confused

• Distance and displacement: Distance is the total path length and is a scalar; displacement is the change in position and is a vector.
• Speed and velocity: Speed is distance per unit time and has no direction; velocity is displacement per unit time and includes direction.
• Speed and acceleration: Acceleration measures change in velocity, not speed; an object accelerates whenever its velocity changes.
• Average and instantaneous velocity: Average velocity applies over a time interval, whereas instantaneous velocity applies at a particular instant.
• Uniform and non-uniform motion: Uniform motion has constant velocity; non-uniform motion has velocity that changes with time.
• Uniform velocity and uniform acceleration: Uniform velocity means velocity is constant; uniform acceleration means velocity changes by equal amounts in equal time intervals.
• Slope and area on graphs: The slope of a position-time graph gives velocity, while the slope of a velocity-time graph gives acceleration; areas under velocity-time and acceleration-time graphs give displacement and change in velocity respectively.
• Zero displacement and zero distance: Returning to the starting point gives zero displacement but non-zero distance.
• Negative velocity and negative speed: Velocity may be negative because of the chosen direction; speed is a scalar and is not negative.
• Constant-acceleration equations and general motion: The equations of motion apply only when acceleration is constant.

What Gets Asked

• Calculate displacement using $\Delta x=x_2-x_1$, identifying the sign from the chosen positive direction. A common error is treating displacement as total distance travelled.
• Calculate average speed and average velocity. Marks are lost by using total distance for velocity or total displacement for speed.
• Determine instantaneous velocity or acceleration using $v=dx/dt$ and $a=dv/dt=d^{2}x/d{t}^2$. These quantities must not be confused with their average forms.
• Select and apply the appropriate constant-acceleration equation. The equations apply only when acceleration is constant, and $u$, $v$, $a$, $t$, and $x$ must be identified correctly.
• Interpret position-time, velocity-time, and acceleration-time graphs. The relevant slope or area must be used: position-time slope gives velocity, velocity-time slope gives acceleration, velocity-time area gives displacement, and acceleration-time area gives change in velocity.
• Analyse vertical motion and free fall using $g\approx 9.8\text{m/s²}$ downward, or approximately $10\text{m/s²}$ for simple calculations. A frequent source of error is failing to apply a consistent sign convention.