Section 1 Motion in One Dimension Chapter 2 Section 1 Motion in One Dimension

Dynamics The branch of physics involving the motion of an object and the relationship between that motion and other physics concepts Kinematics is a part of dynamics In kinematics, you are interested in the description of motion Not concerned with the cause of the motion

One-Dimensional Motion Simplest form of motion Straight-line only Takes place over time Requires a frame of reference

Frame of Reference Defines position Gives a “starting point” for motion Measured against a coordinate axis system x- or y-axis for one-dimensional motion

Quantities in Motion Any motion involves three concepts Displacement Velocity Acceleration These concepts can be used to study objects in motion

Vector and Scalar Quantities Vector quantities need both magnitude (size) and direction to completely describe them Generally denoted by boldfaced type and an arrow over the letter Vector quantities must have a sign (+ or ) Scalar quantities are completely described by magnitude only

Displacement Defined as the change in position ∆x = xf – xi f stands for final and i stands for initial Initial position is sometimes notes as xi May be represented as y if motion is vertical (later) Units are meters (m) in SI, centimeters (cm) in cgs or feet (ft) in US Customary

Displacements

Displacement Isn’t Distance Displacement is a change in position Depends on the point of reference Example: Throw a ball straight up and then catch it at the same point you released it The distance is twice the height The displacement is zero

Speed and Velocity

Speed The average speed of an object is defined as the total distance traveled divided by the total time elapsed Speed is a scalar quantity

Speed, cont Average speed totally ignores any variations in the object’s actual motion during the trip The total distance and the total time are all that is important SI units are m/s

Velocity It takes time for an object to undergo a displacement The average velocity is rate at which the displacement occurs Must include a time interval Velocity is a vector quantity

Average Velocity (See Page 43 of text)

Velocity continued Direction will be the same as the direction of the displacement (time interval is always positive) + or - is sufficient SI unit of velocity is meters per second (m/s) Other units may be given in a problem, but generally will need to be converted to these

Speed is Not Velocity Cars on both paths have the same average velocity since they had the same displacement in the same time interval The car on the blue path will have a greater average speed since the distance it traveled is larger

Speed is Not Velocity Average speed is total distance divided by total time Average velocity is total displacement divided by total time

Graphical Interpretation of Velocity Velocity can be determined from a position-time graph Average velocity equals the slope of the line joining the initial and final positions An object moving with a constant velocity will have a graph that is a straight line

Average Velocity, Constant The straight line indicates constant velocity The slope of the line is the value of the average velocity

Average Velocity, Non Constant The motion is non-constant velocity The average velocity is the slope of the blue line joining two points

Instantaneous Velocity The limit of the average velocity as the time interval becomes infinitesimally short, or as the time interval approaches zero The instantaneous velocity indicates what is happening at every point of time

Instantaneous Velocity on a Graph Instantaneous velocity is the slope of the line tangent to the position-vs.-time graph at any point The instantaneous speed is defined as the magnitude of the instantaneous velocity

Uniform Velocity Uniform velocity is constant velocity The instantaneous velocities are always the same All the instantaneous velocities will also equal the average velocity

Acceleration Section 2 in textbook

Acceleration Changing velocity (non-uniform) means an acceleration is present Acceleration is the rate of change of the velocity Units are m/s² (SI), cm/s² (cgs), and ft/s² (US Cust)

Average Acceleration Vector quantity When the sign of the velocity and the acceleration are the same (either positive or negative), then the speed is increasing When the sign of the velocity and the acceleration are in the opposite directions, the speed is decreasing

Instantaneous and Uniform Acceleration The limit of the average acceleration as the time interval goes to zero When the instantaneous accelerations are always the same, the acceleration will be uniform The instantaneous accelerations will all be equal to the average acceleration

Constant (Uniform) Acceleration We have 3 very useful equations for objects moving in a straight line. For these we will only be concerned with their magnitudes. Variable assignments: t = time of acceleration a = acceleration u = the initial speed/velocity v = final speed/velocity s = distance traveled in time t

Problem-Solving Hints Read the problem Draw a diagram Choose a coordinate system, label initial and final points, indicate a positive direction for velocities and accelerations Label all quantities, be sure all the units are consistent Convert if necessary Choose the appropriate kinematic equation 29

Problem-Solving Hints, cont Solve for the unknowns You may have to solve two equations for two unknowns Check your results Estimate and compare Check units 30

Kinematic Equations Used in situations with uniform acceleration

Displacement with constant acceleration Page 52 in Text This equation applies only when acceleration is constant. Use when you want to find the displacement of any object moving with constant acceleration.

Sample Problem A racing car reaches a speed of 42 m/s. It then begins a uniform negative acceleration, using its parachute and braking system, and comes to rest 5.5 s later. Find the distance that the car travels during braking. Δx= 120m

Sample #2 A bicyclist accelerates from 5.0 m/s to 16 m/s in 8.0 s. Assuming uniform acceleration, what distance does the bicyclist travel during this time interval? 84m

Velocity with constant Acceleration Pg 54 in text Shows velocity as a function of acceleration and time Use to find final velocity of an object after it has accelerated at a constant rate for any time interval

Displacement with Constant Acceleration Use to find displacement of an object moving with constant acceleration Use to find displacement required for an object to reach a certain speed or to come to a stop.

Velocity and Displacement with Constant Acceleration For most situations, you have to use both equations to solve problems.

Sample Problem #1 A plane starting at rest at one end of a runway undergoes a uniform acceleration of 4.8 m/s2 for 15 s before takeoff. What is its speed at takeoff? How long must the runway be for the plane to be able to take off?

Second of the equations Gives velocity as a function of acceleration and displacement Use when you don’t know and aren’t asked for the time

Sample Problem A person pushing a stroller starts from rest, uniformly accelerating at a rate of 0.500 m/s2 . What is the velocity of the stroller after it has traveled 4.75m?

Graphical Interpretation of Acceleration Average acceleration is the slope of the line connecting the initial and final velocities on a velocity-time graph Instantaneous acceleration is the slope of the tangent to the curve of the velocity-time graph

Average Acceleration

Direction of Acceleration Moving right as it speeds up Moving right as it slows down Moving left as it speeds up Moving left as it slows down Vi a +  To make the situations more concrete, use an automobile as an example. For example, the first combination would be a car moving to the right (v is +) and accelerating to the right (a is +), so the speed will increase. Some students may be confused by the latter two examples, thinking that a negative acceleration corresponds to slowing down and a positive acceleration corresponds to speeding up. Emphasize that the directions of velocity and acceleration must both be taken into account. In the third example, the velocity and acceleration are in the same direction, so the object is speeding up. In the fourth case, they are in opposite directions, so the object is slowing down.

Relationship Between Acceleration and Velocity Uniform velocity (shown by red arrows maintaining the same size) Acceleration equals zero

Relationship Between Velocity and Acceleration Velocity and acceleration are in the same direction Acceleration is uniform (blue arrows maintain the same length) Velocity is increasing (red arrows are getting longer) Positive velocity and positive acceleration

Relationship Between Velocity and Acceleration Acceleration and velocity are in opposite directions Acceleration is uniform (blue arrows maintain the same length) Velocity is decreasing (red arrows are getting shorter) Velocity is positive and acceleration is negative

Notes on the equations Gives displacement as a function of velocity and time Use when you don’t know and aren’t asked for the acceleration

Graphical Interpretation of the Equation y = at = v

Free-Fall Chapter 2 Section 3

What do you think? Observe a metal ball being dropped from rest. Describe the motion in words. Sketch a velocity-time graph for this motion. Observe the same ball being tossed vertically upward and returning to the starting point. When asking students to express their ideas, you might try one of the following methods. (1) You could ask them to write their answers in their notebook and then discuss them. (2) You could ask them to first write their ideas and then share them with a small group of 3 or 4 students. At that time you can have each group present their consensus idea. This can be facilitated with the use of whiteboards for the groups. The most important aspect of eliciting student’s ideas is the acceptance of all ideas as valid. Do not correct or judge them. You might want to ask questions to help clarify their answers. You do not want to discourage students from thinking about these questions and just waiting for the correct answer from the teacher. Thank them for sharing their ideas. Misconceptions are common and can be dealt with if they are first expressed in writing and orally. Students may not see that the ball is accelerating uniformly and, as a result, they may draw a graph that rises and then starts to level off. You might also ask them how their answers would change if you dropped a similar but lighter ball. Then, drop the two simultaneously and let them observe the motion.

Free Fall Assumes no air resistance Acceleration is constant for the entire fall Acceleration due to gravity (ag or g ) Has a value of -9.81 m/s2 Negative for downward Roughly equivalent to -22 (mi/h)/s When presenting this slide, you may wish to refer students to Figure 14 in their textbook. You could also begin this slide with a video clip of the “feather and hammer” experiment on the moon. It is available on NASA and other web sites. Perform an internet search with the terms “feather and hammer on moon” to find a link. Discuss the effects of air resistance with students. With air resistance, an object will continue to accelerate at a smaller rate until the acceleration is zero. At that point the object has reached “terminal velocity.”

All previous kinematic equations still work Free Fall All previous kinematic equations still work Since motion is vertical (up or down) x becomes y When presenting this slide, you may wish to refer students to Figure 14 in their textbook. You could also begin this slide with a video clip of the “feather and hammer” experiment on the moon. It is available on NASA and other web sites. Perform an internet search with the terms “feather and hammer on moon” to find a link. Discuss the effects of air resistance with students. With air resistance, an object will continue to accelerate at a smaller rate until the acceleration is zero. At that point the object has reached “terminal velocity.”

Equations

Free Fall For a ball tossed upward, make predictions for the sign of the velocity and acceleration to complete the chart. Velocity (+, -, or zero) Acceleration When halfway up When at the peak When halfway down + - zero When presenting this slide, you may wish to refer students to Figure 15 in their textbook. You can also demonstrate the motion for students. Toss a ball up and catch it. Ask students to focus on the spot half-way up and observe the motion at that time. They can then predict the sign for the velocity and acceleration at that point. Then ask students to focus on the peak and, finally, on a point half-way down. Often students believe the acceleration at the top is zero because the velocity is zero. Point out to them that acceleration is not velocity, but changing velocity. At the top, the velocity is changing from + to -. Ask students to explain each combination above. For example, a positive velocity (moving upward) and a negative acceleration (downward) would cause the velocity to decrease.

Graphing Free Fall Based on your present understanding of free fall, sketch a velocity-time graph for a ball that is tossed upward (assuming no air resistance). Is it a straight line? If so, what is the slope? Compare your predictions to the graph to the right. Now students are asked to graph the motion they just observed. This graph should match the answers to the chart on the last slide. Remind them that they are graphing velocity, but acceleration is the slope of the velocity-time graph. Student graphs may have a different initial velocity and a different x-intercept (the time at which the velocity reaches zero), but their graphs should have the same shape and slope as the one given on the slide. Point out that the velocity is zero at the peak (t = 1.1 s for this graph) while the acceleration is never zero because the slope is always negative. Help them get an approximate slope for the graph shown on the slide. It should be close to -9.81 (m/s)/s.

Classroom Practice Problem Justin hits a volleyball so that it moves with an initial velocity of 6.0 m/s straight upward. If the volleyball starts from 2.0 m above the floor, how long will it be in the air before it strikes the floor? The equations from Section 2 apply because this is uniform acceleration. Simply use “y” instead of “x,” and the acceleration is -9.81 m/s2. Allow students some time to get the answers for t = 1.00 s, and then show them the calculations. Then have them continue with the following rows of the table. Students can use equation (4) from the previous lecture to get y, and the second version of equation (2) to get v. Or, they could get y by using equation (1) after getting the velocity, but they must get the average before using equation (1). Point out to students that the ball turns around between the 2.00 and 3.00 second mark. This makes sense, since it starts with a velocity of 15.2 m/s and loses 9.81 m/s of it’s velocity each second (in other words, the velocity decreases by 9.8 m/s in each step).

Now what do you think? Review the descriptions and graphs you created at the beginning of the presentation. Do you want to make any modifications? For the second graph, circle the point representing the highest point of the toss. Have students revisit their original descriptions and graphs to see if they want to make any modifications based on what they have learned from the presentation. Have a discussion with students about what changes they have made, and ask them to explain why they made the changes.