Motion in One Dimension
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Questions and Answers

Newton's 2nd law states that acceleration is directly proportional to the ______ and inversely proportional to the ______.

net force, mass

For a constant force, acceleration remains constant.

True (A)

What is the relationship between velocity and acceleration when acceleration is constant?

Velocity increases linearly with time.

Which of the following represents the relationship between velocity (v) and time (t) for constant acceleration? (Select all that apply)

<p>v = vi + at (C), v = vi - at (D)</p> Signup and view all the answers

Match the following terms with their corresponding descriptions.

<p>acceleration = rate of change of velocity velocity = rate of change of displacement displacement = change in position time = duration of an event</p> Signup and view all the answers

The area under the velocity-time graph represents the ______.

<p>displacement</p> Signup and view all the answers

How is the slope of the velocity-time graph related to acceleration?

<p>The slope of the velocity-time graph is equal to the acceleration.</p> Signup and view all the answers

The acceleration-time graph for a constant acceleration will be a straight line.

<p>True (A)</p> Signup and view all the answers

Which of the following expressions represents the total displacement of an object with constant acceleration, given its initial velocity (vi), final velocity (vf), and time (t)?

<p>x - xi = (vi + vf) * t / 2 (D)</p> Signup and view all the answers

The formula 'x - xi = vi * t + (1/2) * a * t^2' is applicable only when acceleration is constant.

<p>True (A)</p> Signup and view all the answers

What is the significance of the term (1/2) * a * t^2 in the displacement formula 'x - xi = vi * t + (1/2) * a * t^2'?

<p>This term represents the displacement due to the acceleration of the object.</p> Signup and view all the answers

The equation for finding the total displacement of an object with constant acceleration involves the initial velocity, time, and the ______.

<p>acceleration</p> Signup and view all the answers

Match the following terms with their corresponding descriptions in the context of displacement and constant acceleration.

<p>x – xi = Total displacement vi = Initial velocity a = Acceleration t = Time vf = Final velocity</p> Signup and view all the answers

What is the symbol used for instantaneous velocity?

<p>v (D)</p> Signup and view all the answers

The average velocity for a round trip is always zero.

<p>True (A)</p> Signup and view all the answers

In what situations is acceleration assumed to be constant in PHYS 204?

<p>Nearly all situations, except for spring-mass problems.</p> Signup and view all the answers

The time rate of change in speed is defined as ______.

<p>acceleration</p> Signup and view all the answers

What is the average velocity of an object in the second half of a round trip if it travels a distance of L in the negative x direction?

<p>-L/t2 (B)</p> Signup and view all the answers

The average speed and average velocity of a round trip are always equal.

<p>False (B)</p> Signup and view all the answers

What is the primary condition for an object to experience acceleration?

<p>A force must be present.</p> Signup and view all the answers

The formula v f  vi / t represents the _______ acceleration of an object.

<p>average</p> Signup and view all the answers

The equation x  xi  vi * t + ½ * a * t² can be used to solve problems involving constant acceleration in one dimension.

<p>True (A)</p> Signup and view all the answers

What are the three main kinematic equations that can be used to solve problems in one-dimensional motion?

<p>The three main kinematic equations are:</p> <ol> <li><code>v f  vi  at</code></li> <li><code>x  xi  vi * t + ½ * a * t²</code></li> <li><code>v^2 f  vi^2  2a(x  xi)</code></li> </ol> Signup and view all the answers

In the context of the provided content, what does "derive" mean?

<p>To obtain a formula based on definitions and established rules. (A)</p> Signup and view all the answers

Match the given variables to their corresponding physical quantities:

<p>v f = Final velocity vi = Initial velocity a = Acceleration t = Time x = Displacement x i = Initial position</p> Signup and view all the answers

The setup described in Problem 2-42 involves two thin rods fastened to a circular ring, with one rod being vertical and the other making an angle of _______ with the horizontal.

<p>θ</p> Signup and view all the answers

Describe the motion of the beads in Problem 2-42.

<p>The beads are free to slide without friction along the rods. Their motion is determined by the angle of the rod, the initial position and velocity, and the forces acting on them.</p> Signup and view all the answers

What type of motion does the rod of length L in Problem 2-42 undergo?

<p>A combination of linear and circular motion (C)</p> Signup and view all the answers

Which of the following is a key strategy to approach the problem of determining which bead reaches the bottom first? (Select all that apply)

<p>Connect the given information to the relevant formulas. (B), Identify the unknown quantity in the problem. (C), Visualize the problem by creating a sketch. (D)</p> Signup and view all the answers

The acceleration of the red bead is directly proportional to the sine of the angle theta.

<p>True (A)</p> Signup and view all the answers

What is the formula for the displacement of an object with constant acceleration?

<p>x = v₀t + 1/2at²</p> Signup and view all the answers

The time it takes for the red bead to reach point C is denoted by _____.

<p>tr</p> Signup and view all the answers

Match the following symbols with their corresponding physical quantities:

<p>x = Displacement a = Acceleration t = Time v₀ = Initial Velocity g = Acceleration due to Gravity</p> Signup and view all the answers

Which of the following is a reason why people might find this problem difficult?

<p>The connection between given information and relevant formulas is not immediately obvious. (C)</p> Signup and view all the answers

The acceleration of both beads is the same.

<p>False (B)</p> Signup and view all the answers

What is the relationship between the time taken for the red bead to reach point C and the time taken for the blue bead to reach point C?

<p>The red bead takes longer to reach point C.</p> Signup and view all the answers

What is the relationship between the angles 𝜶 and 𝜷 in the isosceles triangles △AOB and △OCB, as depicted in the provided text?

<p>𝜶 + 𝜷 = 90° (C)</p> Signup and view all the answers

The blue bead and the red bead arrive at the bottom of the incline at the same time.

<p>True (A)</p> Signup and view all the answers

What is the formula used to calculate the time taken by the blue bead to reach the bottom of the incline?

<p>tb2 = 2L / (g sin 𝜃)</p> Signup and view all the answers

The time taken by the red bead to reach the bottom of the incline is represented by ______.

<p>tr2</p> Signup and view all the answers

Match the following variables with their corresponding descriptions in the context of the provided text:

<p>R = Radius of the circle L = Length of the incline g = Acceleration due to gravity 𝜃 = Angle of the incline 𝜶 = Base angle of triangle △AOB 𝜷 = Base angle of triangle △OCB</p> Signup and view all the answers

Explain how the equation tb2 = tr2 is obtained in the text.

<p>The equation is obtained by comparing the time expressions for both beads. The time for the blue bead (tb2) is derived from the acceleration along the incline, while the time for the red bead (tr2) is derived from the uniform circular motion. By relating the length of the incline (L), the radius of the circle (R), and the angle (𝜃), the text demonstrates that tb2 = tr2.</p> Signup and view all the answers

What is the main purpose of introducing the two isosceles triangles △AOB and △OCB in the text?

<p>To demonstrate the relationship between the lengths of the incline and the radius of the circle (B)</p> Signup and view all the answers

The text provides a detailed derivation of the time taken by the red bead to reach the bottom of the incline.

<p>False (B)</p> Signup and view all the answers

Flashcards

Instantaneous Velocity

The limit of velocity as time approaches zero.

Average Velocity

Total displacement divided by total time for the trip.

Average Speed

Total distance traveled divided by total time.

Positive Direction

A defined reference direction for measurement.

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Negative Velocity

Indicates motion in the opposite direction.

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Constant Acceleration

Acceleration that does not change over time.

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Force and Acceleration

Acceleration requires a force, either contact or non-contact.

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Displacement

The change in position from start to finish.

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Newton's 2nd Law

Acceleration (a) is proportional to force and inversely proportional to mass.

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Velocity Function

Velocity (v) as a function of time for constant acceleration: v = vi + at.

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Acceleration Slope

Acceleration (a) is the slope of the velocity (v(t)) vs time (t) graph.

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Change in Velocity

The difference between final and initial velocity over time: Δv = v(t) - vi.

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Displacement and Velocity

Displacement (Δx) is calculated as velocity (v) times change in time (Δt).

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Initial Velocity

The starting velocity of an object before acceleration takes effect.

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Graphical Representation

Acceleration as a function of time is shown as a constant line in a graph.

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Total Displacement

The overall distance an object moves from its initial position, calculated as x - xi.

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Area under v vs t graph

Represents total displacement, comprised of rectangles and triangles.

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Displacement Formula

x - xi = vi * t + (1/2) * a * t^2 for constant acceleration.

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Finding x - xi

Given initial velocity (vi), final velocity (vf), and acceleration (a), use displacement formulas.

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1D Motion Equations

Equations used to describe motion in one dimension with constant acceleration.

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Final Velocity Formula

v_f = v_i + a*t gives the final velocity based on initial velocity and acceleration.

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Average Acceleration

a_avg = (v_f - v_i) / t computes average change in velocity over time.

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Kinematic Equation

v_f^2 = v_i^2 + 2a(x - xi) relates velocities, acceleration, and displacement.

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Deriving Motion Parameters

Process of obtaining one motion parameter using others through equations.

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Motion in Vertical Plane

Movement restricted within a vertical plane, often dealing with forces and angles.

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Bead Release Problem

A scenario where two beads are released from rest at the same time along a path.

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1-Dimensional Motion Formula

The equation x = a t² describes motion along a straight line.

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Visual Cues

Sketches or diagrams that illustrate information visually to aid understanding.

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Identifying Unknowns

The process of determining which variables in a problem need to be solved.

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Backward and Forward Work

A strategy that involves connecting known and unknown variables in a problem.

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Red Bead Equation

An equation relating the time for the red bead to travel a distance under gravity.

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Gravity's Role

The force that pulls objects downward, influencing their motion.

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Angle Impact

Represents how the angle of release affects the motion of the beads.

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Isosceles Triangle

A triangle with at least two equal sides and angles.

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Complementary Angles

Two angles that add up to 90 degrees.

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sin(θ)

The ratio of the length of the opposite side to the hypotenuse in a right triangle.

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Acceleration

The rate of change of velocity over time.

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Maximum Height

The highest point reached by an object in projectile motion.

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Initial Velocity (vi)

The starting speed of an object before it changes due to acceleration.

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Equations of Motion

Mathematical formulas that describe the behavior of moving objects.

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Time of Flight

The duration an object is in the air during its motion.

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Study Notes

Motion in One Dimension

  • Definition of Distance and Displacement:

    • Location/position depends on a frame of reference (e.g., coordinate system).
    • Distance: Total length traveled, regardless of direction.
    • Displacement: Difference between final and initial positions, a directional vector quantity.

  • Distance vs. Displacement:

    • Distance is a scalar (magnitude only).
    • Displacement is a vector (magnitude and direction).

  • Speed and Velocity:

    • Speed: Scalar, average speed = total distance/total time.
    • Velocity: Vector, average velocity = total displacement/total time.
    • Instantaneous velocity: Velocity at a specific moment in time (derivative of displacement over time).

  • Acceleration:

    • Acceleration: Rate of change of velocity over time.
    • Constant acceleration: Acceleration remains the same throughout the motion (a common assumption in introductory physics). Often a = change in velocity / change in time.
    • Acceleration requires a force; without a force, there's no acceleration (Newton's Second Law). Different types of forces can result in different kinds of acceleration relationships through time.
    • Often the relationship between acceleration and time can be constant, but can also change with time or other variables.

Problems and Solutions

  • Problem-solving strategies:

    • Clearly define variables (e.g., initial speed, acceleration).
    • Draw a diagram, and use appropriate notation.
    • Establish a reference direction (e.g., positive/negative, up/down).
    • Choose appropriate equations based on information given, such as Newton's Second Law: a = F/m.
    • Use intuition (e.g., for acceleration-based problems) to predict the direction of motion.

  • Example Problems:

    • Include examples of applying concepts to practical scenarios, including projectile motion (like throwing a stone in the air from a building).

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Related Documents

Note_02 1D motion PHYS 204-04 Mechanics PDF

Description

Explore the fundamental concepts of motion in one dimension, including distance, displacement, speed, velocity, and acceleration. Understand the differences between scalar and vector quantities, and dive into the principles of motion with clear definitions and examples.

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