One Dimensional Motion – Velocity & Acceleration Lecturer: Professor Stephen T. Thornton

Reading Quiz: The red and black lines represent the motion of two cars. Which of the following statements is most true? A)Red and black car have the same acceleration. B)Red car always has a higher speed. C)Black car always has a higher speed. D)Black car always has a lower acceleration. E)Red car always has a lower acceleration.

Answer: E The red car has a constant velocity and zero acceleration. Only the black car has acceleration, and it is positive.

Last time Went through class procedures, WebAssign, etc. Discussed length, mass, and time. Significant figures, scientific notation. Problem solving.

Today Distance, displacement. Average speed and average velocity. Instantaneous velocity Acceleration: average, instantaneous Motion with constant acceleration

Copyright © 2009 Pearson Education, Inc. Reference Frames and Displacement Any measurement of position, distance, or speed must be made with respect to a reference frame. For example, if you are sitting on a train and someone walks down the aisle, the person’s speed with respect to the train is a few miles per hour, at most. The person’s speed with respect to the ground is much higher.

One-Dimensional Coordinate System

Displacement Definition: displacement = change in position = final position – initial position Δx = x f - x i Distance = total length traveled

One-Dimensional Motion Along the x-axis What is the total distance traveled (0-4s)? What is the displacement? Do quiz next.

Conceptual Quiz: What is the total distance traveled? A) - 8 m B) -4 m C) -2 m D) +4 m E) +8 m

Conceptual Quiz: What is the total distance traveled? A) - 8 m B) -4 m C) -2 m D) +4 m E) +8 m

Conceptual Quiz: What is the total displacement? A) - 8 m B) -4 m C) -2 m D) +4 m E) +8 m

A) - 8 m B) -4 m C) -2 m Δx = x f – x i = (-1 m) – (1 m) = -2 m D) +4 m E) +8 m Conceptual Quiz: What is the total displacement?

Average Speed Note that this is always a positive number.

Average Velocity Velocity is different than speed, because velocity is a vector.

One-Dimensional Motion Along the x-axis What is average velocity from 0 to 4 s? direction?

Motion along the x axis represented with an x-versus-t graph (same motion as before)

Average Velocity on an x-Versus-t Graph Look at average from point A to B

Average Velocity on an x-versus-t Graph Now look from 2 s to 3 s

Instantaneous Velocity In this way we can find the velocity at any particular instant of time.

What is the instantaneous velocity at t = 1 s? motion

Constant Velocity Graph Instantaneous velocity is equal to average velocity when velocity is constant.

Graphical Interpretation of Average and Instantaneous Velocity

Observable Universe. Recent findings in astrophysics suggest that the observable Universe can be modeled as a sphere of radius R = 13.7 x 10 9 light-years with an average mass density of about 1 x kg/m 3 where only about 4% of the Universe’s total mass is due to “ordinary” matter (such as protons, neutrons, and electrons). Use this information to estimate the total mass of ordinary matter in the observable Universe.

Airplane Motion. An airplane travels 3100 km at a speed of 720 km/h, and then encounters a tailwind that boosts its speed to 990 km/h for the next 2800 km. What was the total time for the trip? What was the average speed of the plane for this trip?

Acceleration Just like velocity is given by the rate of change of position with respect to time, the acceleration is given by the rate of change of velocity with respect to time. We are still dealing with one- dimensional motion, so vector direction is simple.

We use signs to denote the directions of both velocity and acceleration along a particular axis. x  When v is +, motion is to right.  When v is -, motion is to left.  When motion is to the right, and a is +, then object speeds up (accelerates) to the right (case II).  When motion is to the left, and a is +, then object is slowing down and will eventually turn to the right. (not shown here)

Graphical Interpretation of Average and Instantaneous Acceleration If you don’t remember about tangents, please review!

Average Acceleration We must be very careful with units. What are they? m/s 2

Run physlet er1.html Constant velocity, constant acceleration Play with these physlets at home and try the different cases. Time is too short here.

Instantaneous Acceleration Similarity between velocity and acceleration is clear.

Conceptual Quiz: The graph shows position as a function of time for two trains running on parallel tracks. Which of the following is true? A) At time t B, both trains have the same velocity. B) Both trains speed up all the time. C) Both trains have the same velocity at some time before t B. D) Somewhere on the graph, both trains have the same acceleration. time

Conceptual Quiz: The graph shows position as a function of time for two trains running on parallel tracks. Which of the following is true? A) At time t B, both trains have the same velocity. B) Both trains speed up all the time. C) Both trains have the same velocity at some time before t B. D) Somewhere on the graph, both trains have the same acceleration. time

Motion with Constant Acceleration If the acceleration is constant, then we have where v = v 0 at t = 0. This result is easy to show from our definition of a.

The Average Velocity constant acceleration

Constant acceleration Another important result: Note that I have used v i and v f, which is more general than using v 0 and v, because we may want to find the average between some initial and final position other than v 0 and a general v. Let’s determine some important equations.

Insert our previous result for v av Only for constant acceleration!!!!!

This is a very important equation. It relates the position x to the velocity v as a function of time t. But we also know the relationship between velocity v and acceleration a. It was

We now have all the equations we need to solve constant-acceleration problems.

Make sure you look carefully at the examples in the text. I will not go through those.

You drive for 30 minutes at 30 mi/hr and then for another 30 minutes at 50 mi/hr. What is your average speed for the whole trip? A) more than 40 mi/hr B) equal to 40 mi/hr C) less than 40 mi/hr Conceptual QuizI

You drive for 30 minutes at 30 mi/hr and then for another 30 minutes at 50 mi/hr. What is your average speed for the whole trip? A) more than 40 mi/hr B) equal to 40 mi/hr C) less than 40 mi/hr It is 40 mi/hr in this case. Because the average speed is distance/time and you spend the same amount of time at each speed, then your average speed would indeed be 40 mi/hr. Conceptual Quiz

You drive 4 miles at 30 mi/hr and then another 4 miles at 50 mi/hr. What is your average speed for the whole 8-mile trip? A) more than 40 mi/hr B) equal to 40 mi/hr C) less than 40 mi/hr Conceptual Quiz

You drive 4 miles at 30 mi/hr and then another 4 miles at 50 mi/hr. What is your average speed for the whole 8-mile trip? A) more than 40 mi/hr B) equal to 40 mi/hr C) less than 40 mi/hr It is not 40 mi/hr! Remember that the average speed is distance/time. Because it takes longer to cover 4 miles at the slower speed, you are actually moving at 30 mi/hr for a longer period of time! Therefore, your average speed is closer to 30 mi/hr than it is to 50 mi/hr. Conceptual Quiz