PHYS 2110: Mechanics – Motion with Constant Acceleration and Reference Frames

Motion with Constant Acceleration (1D) and Inertial Frames

Introduction

This study guide covers the foundational concepts of one-dimensional motion with constant acceleration, including the use of vector quantities, kinematic equations, graphical analysis, and the role of reference frames in mechanics. These topics are essential for understanding motion in physics and solving related problems.

Review: Working with Vector Quantities

Displacement, Velocity, Acceleration, and Changes in Velocity

• Vector quantities have both magnitude and direction. Examples include displacement, velocity, and acceleration.

• To describe the direction of a vector, assign a coordinate axis and use positive or negative signs to indicate direction.

• Multiplying a vector by +1 or -1 will "flip" its direction; multiplying by a positive scalar changes only the magnitude.

Example: If a car moves east and east is defined as the positive x-direction, its velocity is positive. If it moves west, its velocity is negative.

Problem Solving with Vectors

Start with the Coordinate Axis

• Define a coordinate axis and decide which direction is positive.

• Make a sketch and add the direction of all relevant vectors.

• Compare the direction of vectors to the coordinate axis to determine signs for the scalar components.

Example: A driver avoids an accident by slamming the brakes, causing the car to slow down. If the acceleration is marked as -6.6 m/s2, the negative sign indicates the car is slowing down. Use the kinematic equation: $v_f^2 = v_i^2 + 2a \Delta x$

Instantaneous Acceleration

Definition and Calculation

• When acceleration is not constant, determine the slope of the tangent to the velocity-time (v(t)) curve at an instant.

• Acceleration is the derivative of velocity with respect to time:

$a(t) = \frac{dv}{dt}$

Instantaneous Acceleration in v(t) Graphs

Graphical Interpretation

• Instantaneous acceleration is the slope of the tangent to the v(t) graph at a given time.

• For non-constant acceleration, divide the motion into small intervals and take the limit as $\Delta t \to 0$.

Change in Position from v(t) Graph

Calculus Approach

• To find the change in position during a time interval, determine the area under the v(t) graph.

• For constant velocity: $\Delta x = v \Delta t$

• For variable velocity: $\Delta x = \int_{t_0}^{t_1} v(t) dt$

• For constant acceleration: $\Delta x = v_0 t + \frac{1}{2} a t^2$

Relationship between Position, Velocity, and Acceleration

Derivatives and Kinematic Equations

• Velocity is the first derivative of position with respect to time:

$v(t) = \frac{dx}{dt}$

• Acceleration is the second derivative of position with respect to time:

$a(t) = \frac{d^2x}{dt^2}$

Motion with Constant Acceleration

Kinematic Equations

• These equations describe motion when acceleration is constant:

Equation	Description
$v_f = v_i + a \Delta t$	Final velocity after time interval
$x_f = x_i + v_i \Delta t + \frac{1}{2} a (\Delta t)^2$	Final position after time interval
$v_f^2 = v_i^2 + 2a \Delta x$	Relates velocity and displacement
$\Delta x = \frac{v_i + v_f}{2} \Delta t$	Displacement using average velocity

Acceleration due to Gravity

Projectile Motion and Gravity

• Near Earth's surface, acceleration due to gravity is $g \approx 9.8 \text{ m/s}^2$ downward.

• "Little g" decreases with greater altitude and varies by location.

• For vertical motion, modify kinematic equations by replacing $a$ with $-g$ (if down is negative) or $+g$ (if down is positive).

Location	g (m/s2)
Anchorage	9.826
Seattle	9.811
Washington DC	9.801

Example: NASA Drop Tower experiments use free fall to study motion under gravity.

Procedure: Solving Problems

Steps for Problem Solving

1. Getting started: Carefully analyze the information, identify what is being asked, and organize data.

2. Devise a plan: Use physical relationships or equations relevant to the problem.

3. Execute plan: Perform calculations and check your work for errors.

4. Evaluate results: Assess if the answer makes sense physically and mathematically.

Motion in front of Windows (Concept Question)

Ranking Motion by Change in Speed and Time

• Rank the windows by the change in speed of the ball as it passes each window.

• Rank the shaded area by the time the ball spends passing each window.

Reference Frames and Relativity of Motion

Definition and Application

• A reference frame is a coordinate system in which an observer makes position measurements.

• Kinematic quantities (position, displacement, velocity, acceleration) depend on the chosen reference frame.

• Compare the motion of objects as seen from different reference frames.

Reference Frame	Observer	Measured Quantity
Earth	Stationary observer	Velocity of car relative to Earth
Moving cart	Observer on cart	Velocity of car relative to cart

Example: Two carts on a track: the measured velocity depends on whether the ruler is fixed to the track or moves with the cart.

What You Should Know

• Qualitative shape of motion graphs for constant acceleration.

• Four equations for motion with constant acceleration.

• Motion along an incline with negligible friction has a constant acceleration of $a = g \sin \theta$.

Additional info: For motion on an incline, the acceleration is less than $g$ and depends on the angle $\theta$ of the incline.