Motion Along a Straight Line: Kinematics in One Dimension

Motion Along a Straight Line

Introduction to Kinematics and Dynamics

Kinematics and dynamics are two fundamental branches of mechanics in physics. Kinematics describes how objects move, while dynamics explains why objects move, focusing on the forces involved. In this section, we focus on one-dimensional (1D) motion, where all movement occurs along a straight line, typically labeled as the x-axis.

• Kinematics: Study of motion without considering its causes.

• Dynamics: Study of the forces that cause motion.

• For 1D motion, only one coordinate (x) is needed to specify position.

Reference Frames and Coordinate Systems

To analyze motion, it is essential to:

• Choose an origin: The point where x = 0.

• Choose a coordinate axis: Typically, the x-axis for 1D motion.

• Positions to the right of the origin are positive; to the left are negative.

Displacement, Velocity, and Speed

Displacement

Displacement is a vector quantity representing the change in position of an object along the x-axis.

• Defined as: $\Delta x = x_2 - x_1$

• Positive if moving in the +x direction, negative if in the -x direction.

• Only the initial and final positions matter, not the path taken.

Position vs. Time Graphs

Plotting position ($x$) versus time ($t$) helps visualize motion. The slope of the line connecting two points on this graph gives the average velocity.

• The graph is a 2D plot, but the motion is still 1D.

• The slope (rise over run) represents velocity.

Average Velocity

Average velocity is the rate of change of displacement with respect to time.

• Formula: $v_{av} = \frac{\Delta x}{\Delta t} = \frac{x_2 - x_1}{t_2 - t_1}$

• Vector quantity; sign indicates direction.

Average Speed

Average speed is the total distance traveled divided by the time interval. It is always positive.

• Formula: $s_{av} = \frac{d}{\Delta t}$

• Scalar quantity; does not indicate direction.

• For any trip, $d \geq |\Delta x|$

Example: Airplane Flight

• Suppose a flight from Houston to Kansas City to Minneapolis covers 1700 km in 4 hours (including layover).

• Average velocity: $v_{av} = \frac{1000\,\text{km}}{4\,\text{h}} = 250\,\text{km/h}$ (direction depends on sign of $\Delta x$)

• Average speed: $s_{av} = \frac{1700\,\text{km}}{4\,\text{h}} = 425\,\text{km/h}$

• Note: Average speed is greater than the magnitude of average velocity if the path is not straight.

Instantaneous Velocity and Acceleration

Instantaneous Velocity

The instantaneous velocity is the velocity of an object at a specific instant. It is the limit of average velocity as the time interval approaches zero.

• Formula: $v_x = \lim_{\Delta t \to 0} \frac{\Delta x}{\Delta t} = \frac{dx}{dt}$

• Graphically, it is the slope of the tangent to the $x$ vs. $t$ curve at a point.

Instantaneous Acceleration

Acceleration is the rate of change of velocity. The instantaneous acceleration is the limit of average acceleration as the time interval approaches zero.

• Average acceleration: $a_{av} = \frac{\Delta v_x}{\Delta t} = \frac{v_{x,2} - v_{x,1}}{t_2 - t_1}$

• Instantaneous acceleration: $a_x = \lim_{\Delta t \to 0} \frac{\Delta v_x}{\Delta t} = \frac{dv_x}{dt} = \frac{d^2x}{dt^2}$

• Graphically, it is the slope of the tangent to the $v_x$ vs. $t$ curve at a point.

Interpreting Graphs

Position vs. Time ($x$ vs. $t$)

• Slope: Gives velocity.

• Concavity: Indicates acceleration (upward = positive, downward = negative).

Velocity vs. Time ($v_x$ vs. $t$)

• Slope: Gives acceleration.

• Area under curve: Gives displacement ($\Delta x$).

Acceleration vs. Time ($a_x$ vs. $t$)

• Area under curve: Gives change in velocity ($\Delta v_x$).

Constant Acceleration: Kinematic Equations

Kinematic Equations for Constant Acceleration

When acceleration is constant, the following equations describe motion:

• $v = v_0 + at$

• $x = x_0 + v_0 t + \frac{1}{2} a t^2$

• $v^2 = v_0^2 + 2a(x - x_0)$

• $x = x_0 + \frac{1}{2}(v_0 + v)t$

Where:

• $x_0$ = initial position

• $v_0$ = initial velocity

• $a$ = constant acceleration

• $t$ = time elapsed

Example: Police Chase

• A car passes a sign at $v_{M0} = 15$ m/s. A police officer starts from rest ($v_{P0} = 0$) and accelerates at $a_P = 3$ m/s$^2$.

• To find when the officer catches up: set $x_P = x_M$ and solve for $t$ using $x_P = \frac{1}{2} a_P t^2$ and $x_M = v_{M0} t$.

• Solution: $t = \frac{2 v_{M0}}{a_P} = \frac{2 \times 15}{3} = 10$ s

• Officer's speed at that time: $v_P = a_P t = 30$ m/s

• Distance from sign: $x = v_{M0} t = 150$ m

Free Fall Motion

Free Fall and Acceleration Due to Gravity

Near Earth's surface, objects in free fall experience a constant acceleration due to gravity, $g = 9.8$ m/s$^2$ (downward). If upward is positive, acceleration is $a = -g$.

• All objects fall with the same acceleration (ignoring air resistance).

• Use kinematic equations with $a = -g$ for vertical motion.

Example: Ball Thrown Upward

• Initial velocity: $v_0 = 15.0$ m/s upward, $y_0 = 0$

• Find position and velocity at $t = 1$ s:

  • $v = v_0 - g t = 15.0 - 9.8 \times 1 = 5.2$ m/s (upward)

  • $y = y_0 + v_0 t - \frac{1}{2} g t^2 = 0 + 15.0 \times 1 - 4.9 \times 1^2 = 10.1$ m

• At $t = 4$ s:

  • $v = 15.0 - 9.8 \times 4 = -24.2$ m/s (downward)

  • $y = 0 + 15.0 \times 4 - 4.9 \times 16 = -18.4$ m (below starting point)

• To find velocity when $y = 5$ m above release:

  • Use $v^2 = v_0^2 - 2g(y - y_0)$

  • Plug in values to solve for $v$

Summary Table: Kinematic Quantities

Quantity	Definition	Graphical Interpretation
Displacement ($\Delta x$)	$x_2 - x_1$	Change in position
Average velocity ($v_{av}$)	$\frac{\Delta x}{\Delta t}$	Slope of secant on $x$ vs. $t$
Instantaneous velocity ($v$)	$\frac{dx}{dt}$	Slope of tangent on $x$ vs. $t$
Average acceleration ($a_{av}$)	$\frac{\Delta v}{\Delta t}$	Slope of secant on $v$ vs. $t$
Instantaneous acceleration ($a$)	$\frac{dv}{dt}$	Slope of tangent on $v$ vs. $t$

Interpreting Areas Under Curves

• Area under $v_x$ vs. $t$ curve: Displacement ($\Delta x$)

• Area under $a_x$ vs. $t$ curve: Change in velocity ($\Delta v_x$)

Key Points

• Displacement and velocity are vector quantities; speed is scalar.

• For constant acceleration, use kinematic equations to solve for unknowns.

• Graphical analysis (slope and area) provides insight into motion.