Kinematics in One Dimension: Acceleration, Motion with Constant Acceleration, and Free Fall

Acceleration in One Dimension

Learning Objectives

• Relate any change in the velocity of a particle to acceleration.

• Apply the equations of kinematics to solve linear motion problems with constant acceleration.

• Interpret the magnitude and direction of acceleration, especially in the context of gravity near Earth's surface.

Acceleration and Changes in Velocity

Acceleration is a fundamental concept in kinematics, describing how the velocity of an object changes over time.

• Definition: Acceleration is the rate of change of velocity.

• Formula:

$a = \frac{\Delta v}{\Delta t}$

• Where $a$ is acceleration, $\Delta v$ is the change in velocity, and $\Delta t$ is the time interval.

Changes in Velocity

When an object's velocity changes, it is accelerating. The direction of the acceleration vector is the same as the direction of the change in velocity.

• Component Form: For one-dimensional motion, acceleration is the change in the velocity component along the axis of motion.

• Formula:

$a_x = \frac{v_{x,2} - v_{x,1}}{t_2 - t_1}$

• The acceleration vector points in the same direction as the change in velocity vector.

Speeding Up or Slowing Down

The sign of acceleration does not by itself determine whether an object is speeding up or slowing down; it depends on the direction of both velocity and acceleration.

• When velocity and acceleration are in the same direction, the object speeds up.

• When velocity and acceleration are in opposite directions, the object slows down.

Acceleration as Curvature in Position-Time Graphs

Acceleration can be visualized as the curvature of a position vs. time ($x(t)$) graph.

• An upward curvature (curve lies above the tangent) indicates positive acceleration.

• A downward curvature (curve lies below the tangent) indicates negative acceleration.

Equations for Motion with Constant Acceleration

Average Velocity for Constant Acceleration

• For constant acceleration, the average velocity over a time interval is:

$v_{\text{avg}} = \frac{v_{x,1} + v_{x,2}}{2}$

• Where $v_{x,1}$ is the initial velocity and $v_{x,2}$ is the final velocity.

Deriving the Position Equation

• Starting from the definition of velocity and integrating for constant acceleration:

$x_2 = x_1 + v_{x,1} \Delta t + \frac{1}{2} a_x (\Delta t)^2$

• This is a second-degree polynomial in time, describing position as a function of time for constant acceleration.

Useful Kinematic Equations

• Combining the equations for velocity and position, we obtain:

$v_{x,2}^2 = v_{x,1}^2 + 2a_x (x_2 - x_1)$

• This equation relates velocity and displacement without requiring time.

Kinematic Equations Summary

Equation	Variables
$v_{x,2} = v_{x,1} + a_x \Delta t$	Final velocity, initial velocity, acceleration, time
$x_2 = x_1 + v_{x,1} \Delta t + \frac{1}{2} a_x (\Delta t)^2$	Final position, initial position, initial velocity, acceleration, time
$v_{x,2}^2 = v_{x,1}^2 + 2a_x (x_2 - x_1)$	Final velocity, initial velocity, acceleration, displacement
$x_2 = x_1 + \frac{1}{2}(v_{x,1} + v_{x,2})\Delta t$	Final position, initial position, initial and final velocity, time

Stopping Distance

Real-Life Scenario

Stopping distance is the distance required for a vehicle to come to a complete stop after the brakes are applied.

• It depends on the initial speed and the magnitude of the (negative) acceleration (deceleration).

• For a given deceleration $a$, the stopping distance $d$ from initial speed $v_0$ is:

$d = \frac{v_0^2}{2|a|}$

• Stopping distance increases with the square of the initial speed.

• Doubling the speed increases the stopping distance by a factor of four.

Acceleration Due to Gravity and Free Fall

Acceleration Due to Gravity

• All objects in free fall near Earth's surface experience the same acceleration due to gravity, regardless of mass (ignoring air resistance).

• The standard value is:

$g = 9.8\ \text{m/s}^2$

• Direction: Downward, toward the center of the Earth.

Free Fall

• Definition: The motion of an object under the influence of gravity only is called free fall.

• In the absence of air resistance, all objects fall with the same acceleration $g$.

• Position and velocity as functions of time (if dropped from rest):

$v = -gt$

$y = y_0 - \frac{1}{2}gt^2$

• Where $y_0$ is the initial height, $v$ is the velocity at time $t$ (downward is negative).

Experimental Evidence

• Galileo's experiments and modern demonstrations (e.g., dropping a feather and a hammer in a vacuum) confirm that, in the absence of air resistance, all objects fall at the same rate.

• In air, lighter objects (like feathers) fall more slowly due to air resistance.

Summary Table: Free Fall vs. Air Resistance

Condition	Acceleration	Effect on Objects
In Vacuum	$g = 9.8\ \text{m/s}^2$	All objects fall at same rate
In Air	< $g$ (varies)	Lighter objects fall slower due to air resistance

Key Concepts and Applications

• Constant Acceleration: Many real-world motions (e.g., vehicles braking, objects in free fall) can be modeled using constant acceleration equations.

• Graphical Analysis: Position-time and velocity-time graphs provide visual insight into acceleration and motion characteristics.

• Safety Implications: Understanding stopping distance is crucial for safe driving and engineering design.

• Universal Acceleration: The acceleration due to gravity is a universal constant near Earth's surface, foundational for further studies in physics.