Chapter 2 Motion in One Dimension Flashcards

Chapter 2: Motion in One Dimension

Chapter Goal

• Describe and analyze linear motion.

Preview: Uniform Motion

• Successive images of a rider are the same distance apart, indicating constant velocity.
• Uniform motion involves describing motion using quantities like distance and velocity.

Preview: Acceleration

• Acceleration refers to a rapid change in speed.
• The concept of acceleration helps solve problems involving changing velocities, such as races or a cheetah chasing prey.

Preview: Free Fall

• Free fall occurs when an object's motion (up and down) is determined by gravity alone.
• Problems involve determining how long it takes for an object to go up and come back down.

Preview: Motion Diagrams

• Analyzing motion begins with drawing a motion diagram, marking positions at successive times.
• Creating motion diagrams helps solve problems.

Stop to Think

• A bicycle moves to the left with increasing speed. Selecting the correct motion diagram is important.

Describing Motion

• Use of various symbols and notations for different contexts (e.g., regular parentheses, square brackets, British talk, animorph talk, sarcasm).

Representing Position

• An x-axis is used for horizontal motion and motion on a ramp (positive end to the right).
• A y-axis is used for vertical motion (positive end up).
• Motion diagrams represent position at particular times.
• Tables quantify motion, and graphs display motion as x versus t.
• Example data:
  • Time t (min): 0, 1, 2, 3, 4, 5, 6, 7, 8, 9
  • Position x (m): 0, 60, 120, 180, 200, 220, 240, 340, 440, 540

From Position to Velocity

• Faster speed corresponds to a steeper slope on a position-versus-time graph.
• The slope of a position-versus-time graph represents the object's velocity.

Tactics Box 2.1: Interpreting Position-Versus-Time Graphs

• Determine an object's position at time t by reading the graph at that instant.
• Determine the object's velocity at time t by finding the slope of the position graph at that point; steeper slopes indicate faster speeds.
• Determine the direction of motion by noting the sign of the slope.
  • Positive slopes correspond to positive velocities and motion to the right (or up).
  • Negative slopes correspond to negative velocities and motion to the left (or down).

From Position to Velocity

• A velocity-versus-time graph can be deduced from a position-versus-time graph, representing an object's motion.

QuickCheck Examples (2.1, 2.2, 2.3)

• Matching motion diagrams to corresponding velocity-versus-time graphs.

Example: Finding a Car’s Velocity from Its Position Graph

• Drawing a velocity-versus-time graph from a given position-versus-time graph.
• Describing the car’s motion in words.

From Velocity to Position

• A position-versus-time graph can be deduced from a velocity-versus-time graph.
• The sign of the velocity indicates whether the slope of the position graph is positive or negative.
• The magnitude of the velocity indicates how steep the slope is.

QuickCheck Examples (2.4, 2.5, 2.6)

• Matching motion diagrams or position graphs to corresponding velocity-versus-time graphs.

Uniform Motion

• Straight-line motion with equal displacements during any successive equal-time intervals is uniform motion or constant-velocity motion.
• An object’s motion is uniform if and only if its position-versus-time graph is a straight line.

Equations of Uniform Motion

• The velocity of an object in uniform motion tells us the amount by which its position changes during each second.
• Position equation for an object in uniform motion:
  • x = x0 + vx \Delta t

Mathematical Relationships

• Graphs with the same overall appearance may relate to different physical phenomena but share the same mathematical relationship.

QuickCheck 2.7

• Determining an object's velocity at a specific time using a position graph.

Example Problem

• A soccer player kicks a ball toward the goal, and a defender tries to block it.
• Problem involves calculating the time the defender has to move into position.

From Velocity to Position, One More Time

• The displacement is equal to the area under the velocity graph during the time interval.

QuickCheck 2.8

• Determining the object’s position at a specific time using a velocity graph.
• Displacement = area under the curve.

Instantaneous Velocity

• An object changing its velocity is either speeding up or slowing down.
• An object’s velocity at a specific instant of time t is its instantaneous velocity; "velocity" generally means instantaneous velocity.

Finding the Instantaneous Velocity

• If the velocity changes, the position graph is a curved line.
• Compute a slope at a point by considering a small segment of the graph.

Finding the Instantaneous Velocity

• In a magnified segment of the position graph, the curve appears to be a line segment.
• The slope can be calculated as rise over run; this is the slope at time t and thus the velocity at this instant.

Finding the Instantaneous Velocity

• Graphically, the slope of the curve at a point is the same as the slope of a straight line tangent to the curve at that point.
• Calculating rise over run for the tangent line gives the instantaneous velocity at that instant of time.

Instantaneous Velocity

• Even when the speed varies, the velocity-versus-time graph can determine displacement.
• The area under the curve in a velocity-versus-time graph equals the displacement, even for non-uniform motion.

QuickCheck Examples (2.9, 2.10, 2.11)

• Relating slope on a position-versus-time graph to object's velocity.
• Determining when objects have the same velocity based on their position graphs.

Example: Calculating the Displacement of a Car during a Rapid Start

• Calculating the displacement of a car using its velocity-versus-time graph.

QuickCheck 2.12

• Analyzing a car's motion based on its position-versus-time graph, identifying points where displacement is zero, speed is zero, speed is increasing, and speed is decreasing.

Acceleration

• Acceleration describes an object whose velocity is changing.
• The ratio \Delta v_x / \Delta t is the rate of change of velocity.
• The ratio \Delta v_x / \Delta t is the slope of a velocity-versus-time graph.
• Definition of acceleration as the rate of change of velocity: ax = \Delta vx / \Delta t

Units of Acceleration

• SI unit of velocity: 60 miles per hour = 27 m/s.
• The Corvette speeds up 27 m/s in 3.6 seconds.
• Every second, the Corvette’s velocity changes by 7.5 m/s.
• Acceleration units are abbreviated as “meters per second squared” (m/s^2).
• Table 2.2: Performance data for vehicles
  • 2016 Chevy Corvette: 0 to 60 miles per hour in 3.6 seconds
  • 2016 Chevy Sonic: 0 to 60 miles per hour in 9.0 seconds

Example: Animal Acceleration

• Lions can sustain an acceleration of 8.0 m/s^2 for up to one second.
• Problem involves calculating how much time it takes a lion to reach a typical recreational runner’s top speed of 10 miles per hour.

Representing Acceleration

• An object’s acceleration is the slope of its velocity-versus-time graph.

Representing Acceleration

• We can find an acceleration graph from a velocity graph.

QuickCheck Examples (2.13, 2.14)

• Determining velocity vector and signs of position and velocity for a particle.

Example Problem

• A ball moves to the right across a ramp.
• Sketching graphs of velocity versus time and acceleration versus time.

The Sign of the Acceleration

• An object can move right or left (or up or down) while either speeding up or slowing down.
• Whether an object that is slowing down has a negative acceleration depends on the direction of motion.
  • Object moving to the right and speeding up: vx > 0, ax > 0
  • Object moving to the left and slowing down: vx < 0, ax > 0

The Sign of the Acceleration

• An object can move right or left (or up or down) while either speeding up or slowing down.
• Whether an object that is slowing down has a negative acceleration depends on the direction of motion.
  • Object moving to the right and slowing down: vx > 0, ax < 0
  • Object moving to the left and speeding up: vx < 0, ax < 0

QuickCheck Examples (2.15 - 2.25)

• Various scenarios involving motion diagrams, cyclists, carts, and graphs to determine acceleration, velocity, and position relationships.

Motion with Constant Acceleration

• The slope of the graph in the velocity graph determines the acceleration of the rocket.

Constant Acceleration Equations

• Use acceleration to find velocity at a later time:
  • Velocity equation for an object with constant acceleration: vf = vi + a \Delta t

Constant Acceleration Equations

• The velocity-versus-time graph for constant-acceleration motion is a straight line with value v{0x} at time t0 and slope a_x.
• The displacement during a time interval is the area under the velocity-versus-time graph.

Constant Acceleration Equations

• The shaded area can be subdivided into a rectangle and a triangle. Adding these areas gives:
  • Position equation for an object with constant acceleration: x = x0 + v0 \Delta t + (1/2) a (\Delta t)^2

Constant Acceleration Equations

• Combining the velocity and position equations for constant acceleration:
  • vf^2 = vi^2 + 2 a (\Delta x)
  • \Delta x is the displacement (not the distance!).

Constant Acceleration Equations

• Velocity changes steadily: v = v_0 + at
• The position changes as the square of the time interval: x = x0 + v0t + (1/2)at^2
• Change in velocity in terms of distance: v^2 = v_0^2 + 2a\Delta x

Example: Coming to a Stop in a Car

• Driving at 15 m/s and stopping in 1.5 seconds.
• Problem involves finding the distance the car travels while braking.

Example Problem: Reaching New Heights

• Spud Webb's vertical leap: 110 cm.
• To jump this high, determine the speed at which he would leave the ground.

Example: Finding the Displacement of a Drag Racer

• A drag racer travels 6.0 meters in 1.0 second from rest.
• Suppose the car continues this acceleration for an additional 4.0 seconds; how far will the car be from the starting line?

Problem-Solving Approach

• Break down complex problems into smaller steps.
• Four steps: Strategize, Prepare, Solve, and Assess.

Problem-Solving Approach

• Strategize: Big-picture questions about the problem.
  • What kind of problem is this?
  • What’s the correct general approach?
  • What should the answer look like?

Problem-Solving Approach

• Prepare: Identify important elements and collect information.
  • Drawing a picture.
  • Collecting necessary information.
  • Doing preliminary calculations.

Problem-Solving Approach

• Solve: Perform the mathematics or reasoning to arrive at the answer.

Problem-Solving Approach

• Assess: Check if the answer makes sense.
  • Does the solution answer the question asked?
  • Does the answer have the correct units and number of significant figures?
  • Does the computed value make physical sense?
  • Can you estimate what the answer should be to check the solution?
  • Does the final solution make sense in the context of the material being learned?

Example: Kinematics of a Rocket Launch

• A Saturn V rocket launched straight up with constant acceleration of 20 m/s^2.
• After 150 s, how fast is the rocket moving, and how far has it traveled?

Example: Calculating the Minimum Length of a Runway

• A fully loaded Boeing 747 accelerates at 2.3 m/s^2.
• Its minimum takeoff speed is 70 m/s.
  • How much time will the plane take to reach its takeoff speed?
  • What minimum length of runway does the plane require for takeoff?

Example: Calculating the Minimum Length of a Runway

• Set x0 and t0 equal to zero at the starting point of the motion when the plane is at rest and the acceleration begins.

Example Problem: Champion Jumper

• A springbok jumps straight up into the air (pronk).
• The speed when leaving the ground can be as high as 7.0 m/s.
  • How much time will it take to reach its highest point?
  • How long will it stay in the air?
  • When it returns to earth, how fast will it be moving?

Free Fall

• Free fall occurs when an object moves under the influence of gravity only.
• All objects in free fall have the same acceleration, regardless of their mass.
• Air resistance is ignored.

Free Fall

• Apollo 15 astronaut David Scott dropped a hammer and a feather on the moon simultaneously; both hit the ground at the same time.

Free Fall

• The motion diagram for an object released from rest and falling freely is the same for all falling objects.

Free Fall

• The free-fall acceleration always points downward.
• Any object moving under the influence of gravity only is in free fall.
• The average value of free-fall acceleration on Earth is approximately 9.8 m/s^2
  Class Video: Free Fall

Free Fall

• g is always positive.
• Even though a falling object speeds up, it has negative acceleration.
• Kinematic equations for constant acceleration can be used for free fall.
• g is the free-fall acceleration; other planets have different values of g.

QuickCheck Examples (2.26, 2.27, 2.28)

• Analyzing the acceleration and velocity of an arrow launched vertically upward.

Example: Analyzing a Rock’s Fall

• A heavy rock is dropped from rest at the top of a cliff and falls 100 m before hitting the ground.
  • How long does the rock take to fall to the ground?
  • What is its velocity when it hits?

Example: Finding the Height of a Leap

• A springbok goes into a crouch and extends its legs forcefully, accelerating at 35 m/s^2 as its legs straighten.
  • At what speed does the springbok leave the ground?
  • How high does it go?

Example Problem

• Passengers on the Giant Drop at Six Flags experience 2.6 seconds of free fall.
  • How fast are the passengers moving at the end of this speeding up phase?
  • If the cars then come to rest in 1.0 seconds, what is the acceleration (magnitude and direction) of this slowing down phase?
  • What is the minimum possible height of the tower?