Study Notes: Acceleration and Motion in One Dimension (Physics with Calculus)

Describing Motion

Acceleration: Definition and Properties

Acceleration is a fundamental concept in physics, representing the rate at which an object's velocity changes over time. It is a vector quantity, meaning it has both magnitude and direction.

• Definition: Acceleration is the change in velocity per unit time.

• Formula: $a = \frac{\Delta v}{\Delta t}$, where $a$ is acceleration, $\Delta v$ is the change in velocity, and $\Delta t$ is the change in time.

• Vector Quantity: Acceleration has both magnitude and direction.

• Graphical Interpretation:

  • The slope of a velocity-time graph represents acceleration.

  • The area under an acceleration-time graph gives the change in velocity over that interval.

Zero Acceleration

Zero acceleration occurs when an object's velocity remains constant over time. In this case, there is no change in speed or direction.

• Key Point: If $a = 0$, then $v$ is constant.

• Example: A car moving at a steady speed of 60 km/h on a straight road has zero acceleration.

Graphing Acceleration

Graphs are useful tools for visualizing acceleration and its effects on motion.

• Velocity-Time Graph: The slope indicates acceleration.

• Acceleration-Time Graph: The area under the curve represents the change in velocity.

• Example: A fan cart accelerating on a track will show a straight, sloped line on a velocity-time graph if acceleration is constant.

Constant Acceleration

Constant acceleration refers to situations where the acceleration does not change over a given time interval. This allows the use of specific kinematic equations to describe motion.

• Key Point: The following equations are valid only when acceleration is constant.

• Kinematic Equations:

  • $v = v_0 + a t$

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

  • $v^2 = v_0^2 + 2 a (x - x_0)$

• Application: These equations are used for objects falling or being thrown near Earth's surface, or for vehicles accelerating on a road.

• Careful: If acceleration changes, apply the equations separately for each interval of constant acceleration.

Constant Acceleration Examples

Examples help illustrate the use of kinematic equations in real-world scenarios.

• Boeing 747 Takeoff:

  • Given: $a = 2.6\ \text{m/s}^2$, $v_0 = 0$, $v = 70\ \text{m/s}$

  • Find: Time to reach takeoff speed and minimum runway length.

  • Solution:

    • Time: $t = \frac{v - v_0}{a} = \frac{70}{2.6} \approx 26.9\ \text{s}$

    • Distance: $x = x_0 + v_0 t + \frac{1}{2} a t^2 = 0 + 0 + \frac{1}{2} \times 2.6 \times (26.9)^2 \approx 940\ \text{m}$

• Fugitive and Freight Train:

  • Train speed: $v_{train} = 5\ \text{m/s}$

  • Fugitive starts from rest, $a = 1.4\ \text{m/s}^2$

  • Find: Time to catch up.

  • Solution:

    • Set positions equal: $x_{fugitive} = x_{train}$

    • $x_{fugitive} = \frac{1}{2} a t^2$

    • $x_{train} = v_{train} t$

    • Equate: $\frac{1}{2} a t^2 = v_{train} t$

    • Solve for $t$: $t = \frac{2 v_{train}}{a} = \frac{2 \times 5}{1.4} \approx 7.14\ \text{s}$

• Fugitive with Maximum Speed:

  • Fugitive accelerates to $6.0\ \text{m/s}$, then maintains that speed.

  • Requires splitting the problem into two intervals: acceleration phase and constant speed phase.

  • Additional info: Use kinematic equations for each phase and sum the times.

Free Fall

Free fall describes the motion of objects under the influence of gravity alone, ignoring air resistance. The acceleration due to gravity is constant and directed downward.

• Acceleration due to Gravity: $g = 9.8\ \text{m/s}^2$ (downward)

• Key Point: Applies whether the object is moving upward or downward.

• Formulas:

  • $v = v_0 + g t$

  • $y = y_0 + v_0 t + \frac{1}{2} g t^2$

• Example: A ball thrown downward from a height of $20\ \text{m}$ with $v_0 = 3\ \text{m/s}$.

• Solution Steps:

  • Find time in air: $y = y_0 + v_0 t + \frac{1}{2} g t^2$; solve for $t$ when $y = 0$.

  • Find velocity at impact: $v = v_0 + g t$

• Example: Ball thrown upward, hits ground after $3\ \text{s}$.

  • Find initial velocity, maximum height, and time to reach $23\ \text{m}$ above ground.

  • Additional info: Use kinematic equations and solve for unknowns using given time and height.

Summary Table: Kinematic Equations for Constant Acceleration

Equation	Variables	Use
$v = v_0 + a t$	Final velocity, initial velocity, acceleration, time	Find final velocity after time $t$
$x = x_0 + v_0 t + \frac{1}{2} a t^2$	Position, initial position, initial velocity, acceleration, time	Find position after time $t$
$v^2 = v_0^2 + 2 a (x - x_0)$	Final velocity, initial velocity, acceleration, displacement	Find final velocity or displacement without time

Additional info: For problems involving multiple intervals of acceleration, apply the kinematic equations separately for each interval and sum the results as needed.