Chapter 3: Acceleration – Structured Study Notes

Acceleration

Changes in Velocity

Acceleration is a fundamental concept in physics describing how an object's velocity changes over time. If an object's velocity is changing, it is said to be accelerating. Acceleration can occur as an object speeds up, slows down, or changes direction. It is a vector quantity, requiring both magnitude and direction for complete specification.

• Definition: The average acceleration in the x direction is given by $a_{x,\text{av}} = \frac{v_{x,f} - v_{x,i}}{t_f - t_i}$.

• Units: The SI unit is meters per second squared (m/s2).

• Direction: Acceleration points in the direction of the change in velocity ($\Delta \vec{v}$).

• Speeding Up vs. Slowing Down: If velocity and acceleration vectors point in the same direction, the object speeds up. If they point in opposite directions, the object slows down.

Example: A sports car increases its speed from 0 to 10 m/s in 2.5 s. Its acceleration is $\frac{10\,\text{m/s} - 0}{2.5\,\text{s}} = 4.0\,\text{m/s}^2$.

Position-Time and Velocity-Time Graphs

Position-versus-time ($x(t)$) and velocity-versus-time ($v_x(t)$) graphs are useful for visualizing acceleration:

• Curvature: Upward curvature in $x(t)$ indicates positive acceleration; downward curvature indicates negative acceleration.

• Slope: The slope of $x(t)$ gives velocity; the slope of $v_x(t)$ gives acceleration.

Acceleration Due to Gravity

Objects falling toward Earth experience acceleration due to gravity, denoted by $g$. Near Earth's surface, $g \approx 9.8\,\text{m/s}^2$ downward. In the absence of air resistance, all objects fall with the same acceleration regardless of their mass or composition.

• Free Fall: Motion under gravity alone is called free fall.

• Galileo's Experiment: Demonstrated that, in vacuum, all objects fall at the same rate.

• Gravity Map: The value of $g$ varies slightly across Earth's surface due to local variations in mass distribution.

Additional info: The gravity map visually represents small variations in $g$ across Earth's oceans, which are important in geophysics and satellite measurements.

Projectile Motion

Projectile motion describes the path of an object launched into the air and influenced only by gravity after launch. The trajectory is typically a parabola.

• Initial Velocity: The launch affects only the initial velocity; subsequent motion is determined by gravity.

• Acceleration: The acceleration remains constant and downward throughout the motion.

• Symmetry: The time to rise to the highest point equals the time to fall back down.

Example: A ball thrown upward with an initial velocity of 8.0 m/s rises to a height of $\frac{(8.0\,\text{m/s})^2}{2 \times 9.8\,\text{m/s}^2} = 3.3\,\text{m}$ and takes 1.6 s for the round trip.

Motion Diagrams

Motion diagrams visually represent the position of an object at equally spaced time intervals. They help organize information about initial and final conditions, velocity, and acceleration.

• Dots: Represent positions at equal time intervals.

• Spacing: Increasing spacing indicates speeding up; decreasing spacing indicates slowing down.

• Procedure: Specify initial and final conditions, indicate acceleration, and label unknowns.

Motion with Constant Acceleration

For constant acceleration, the following equations describe the motion:

• Velocity: $v_{x,f} = v_{x,i} + a_x \Delta t$

• Position: $x_f = x_i + v_{x,i} \Delta t + \frac{1}{2} a_x (\Delta t)^2$

• Velocity squared: $v_{x,f}^2 = v_{x,i}^2 + 2 a_x \Delta x$

• Position as a function of time: $x(t) = x_i + v_{x,i} t + \frac{1}{2} a_x t^2$

• Velocity as a function of time: $v_x(t) = v_{x,i} + a_x t$

Free-Fall Equations

For objects in free fall (upward-pointing x axis):

• $x_f = x_i - \frac{1}{2} g t_f^2$

• $v_{x,f} = -g t_f$

Example: A ball dropped from 20 m takes $t_f = \sqrt{\frac{2h}{g}} = 2.0\,\text{s}$ to hit the ground and has a velocity of $-20\,\text{m/s}$ just before impact.

Inclined Planes

Galileo used inclined planes to study acceleration. For an object rolling down an incline:

• Distance-Time Ratio: $\frac{x}{t^2}$ is constant for constant acceleration.

• Acceleration along incline: $a_x = g \sin \theta$ (where $\theta$ is the angle of incline).

Instantaneous Acceleration

Instantaneous acceleration is the acceleration at a specific moment, defined mathematically as:

• $a_x = \frac{dv_x}{dt}$

• $a_x = \frac{d^2 x}{dt^2}$

For non-constant acceleration, calculus (integration and differentiation) is used to relate position, velocity, and acceleration.

Summary Table: Types of Motion

Type	Position vs. Time	Velocity vs. Time	Acceleration vs. Time
At Rest	Horizontal line	Zero	Zero
Constant Velocity	Straight line	Horizontal line	Zero
Constant Acceleration	Parabola	Straight line	Horizontal line

Key Formulas

• $a_{x,\text{av}} = \frac{v_{x,f} - v_{x,i}}{t_f - t_i}$

• $a_x = \frac{dv_x}{dt} = \frac{d^2 x}{dt^2}$

• $\Delta v_x = \int_{t_i}^{t_f} a_x(t) dt$

• $\Delta x = \int_{t_i}^{t_f} v_x(t) dt$

• $a_x = g \sin \theta$ (inclined plane)

• $g = 9.8\,\text{m/s}^2$ (near Earth's surface)

Examples and Applications

• Free fall: All objects accelerate downward at $g$ regardless of mass.

• Projectile motion: The trajectory is a parabola; time up equals time down.

• Inclined planes: Used to study constant acceleration at reduced rates.

Additional info:

These notes cover the main concepts, equations, and applications of acceleration, including free fall, projectile motion, and motion diagrams. Calculus is introduced for non-constant acceleration, and the relationship between position, velocity, and acceleration is emphasized.