Velocity and Speed Velocity is defined as the rate of change of position. Velocity is a vector (it has direction), so we use the letter v. Remember that speed is the scalar counterpart to velocity. Since it has no direction we use the letter s.

Velocity and Speed Question 1: a) What does the device shown in the picture measure – velocity or speed?

Velocity and Speed Answer 1: This measures speed. It is in fact called a speedometer.

Velocity and Speed Question 1 (cont.): b) What would you have to add to make this device into a velocitometer?

Velocity and Speed Answer 1 (cont.): b) A compass. The dial reads the speed or size of the velocity and the compass reads the direction.

Velocity and Speed A “rate” is how fast something changes over time. In the case of velocity, it is the position that changes. In Algebra class, you may have learned the equation: Distance = Rate x Time Many things can change with time, so in this case, it would be more precise to say “Speed” instead of “Rate”.

Velocity and Speed Velocity is defined as the rate of change of position. Mathematically, we can write this as: This equation states that the velocity is defined as the change in position divided by the time interval or change in time. xf xi - tf ti v Dx Dt =

Velocity and Speed Most often, our initial time will be set to be zero, and we will abbreviate the equation as: Remember, though, that t represents a time interval. v Dx Dt = xf xi - tf ti t

Velocity and Speed Speed is defined mathematically as: Speed is a scalar and, like distance, has no direction and cannot be negative. Distance Time t d = s

Velocity and Speed - Question 2: Check with your neighbor – State two common units of speed or velocity and one uncommon or unusual unit. Distance Time t d = v xf xi - tf ti Dx Dt

Velocity and Speed Answer 2: There are many possible units, as long as they consist of a length unit divided by a time unit. For instance, car speeds are measured in miles per hour (m.p.h.). The word “per” designates division, so m.p.h. could be written as (miles) / (hours). Distance Time t d = s xf xi - tf ti v Dx Dt

Velocity and Speed Answer 2 (cont.): The units match the equation. x and d are measured in length units, such as miles, and t is measured in time units, such as hours. Distance Time hour miles = s v Dx Dt

Velocity and Speed Answer 2 (cont.): Other velocity or speed units might include: Kilometer / hour (car speeds in Europe) Meters / second (lab measurements) Feet per second Inches / century (tectonic plate movement) Kilometers / microsecond (speed of light) v Dx Dt = km Distance Time m s = = hr s

Velocity Measurement Demo Question 3: Watch the object below. Can you determine the value of its velocity by eye? What do you need to measure the velocity? (click to start motion)

Velocity Measurement Demo Answer 3: In order to measure velocity, you will need to measure displacement and time. To measure displacement, we will use the number line scale that we worked with earlier. - + 1 2 3 4 5 -5 -4 -3 -2 -1 X (meters)

Velocity Measurement Demo Answer 3: In order to measure time, you may either use your own stopwatch or watch the clock on the screen which measures in seconds. t (seconds) 12 11 1 2 10 9 3 8 4 7 5 6 - + 1 2 3 4 5 -5 -4 -3 -2 -1 X (meters)

Velocity Measurement Demo You will need to know when to start and stop timing. There will be a countdown: 3, 2, 1, Go. The object will start moving on “Go”. It will continue until it reaches -4 m. Then you will see the word “Stop.” (turn on sound) t (seconds) 12 11 1 2 10 9 3 8 4 7 5 6 - + 1 2 3 4 5 -5 -4 -3 -2 -1 X (meters)

Velocity Measurement Demo Ready? t (seconds) (click to start experiment) 3 2 1 Go 12 11 1 2 10 Stop 9 3 8 4 7 5 6 - + 1 2 3 4 5 -5 -4 -3 -2 -1 X (meters)

Velocity Measurement Demo In case you missed it, we will do it one more time. t (seconds) 12 11 1 2 10 9 3 8 4 7 5 6 - + 1 2 3 4 5 -5 -4 -3 -2 -1 X (meters)

Velocity Measurement Demo Ready? t (seconds) (click to start experiment) 3 2 1 Go 12 11 1 2 10 Stop 9 3 8 4 7 5 6 - + 1 2 3 4 5 -5 -4 -3 -2 -1 X (meters)

Velocity Measurement Demo Question 4: Determine the speed and velocity of the particle. 12 1 2 3 4 5 6 7 8 9 10 11 t (seconds) - + -5 -4 -3 -2 -1 1 2 3 4 5 X (meters)

Velocity Measurement Demo Answer 4: The speed of the particle is 1.2 m/s. s = d/t = 6m / 5s = 1.2 m/s t (seconds) 12 11 1 2 10 9 3 8 4 7 5 6 - + -5 -4 -3 -2 -1 1 2 3 4 5 X (meters)

Velocity Measurement Demo Answer 4 (cont.): The velocity is -1.2 m/s. t (seconds) 12 v xf xi - tf ti = -4m 2m 5s 0s -6m -1.2 m/s 11 1 2 10 9 3 8 4 7 5 6 - + -5 -4 -3 -2 -1 1 2 3 4 5 X (meters)

Velocity Measurement Demo Answer 4 (cont.): The negative sign in the velocity represents the direction, in this case to the left. The direction an object moves is the direction of its velocity. t (seconds) 12 11 1 2 10 9 3 8 4 7 5 6 - + -5 -4 -3 -2 -1 1 2 3 4 5 X (meters)

Velocity Measurement Demo Police officers will sometimes measure automobile speeds in just this way. In the VASCAR system, officers measure the time it takes a vehicle to move from one position to another. In the picture, the circled marks denote the start and end positions.

Velocity Measurement Demo Knowing the distance and the time, the speed can be calculated. Unlike RADAR, which requires the officer to locate by the side of the road, VASCAR speed measurements can be done from hilltops or the air.

Average, Instantaneous, and Uniform We can characterize velocity in different ways, depending on how it is measured. Average velocity (which you are most familiar with) is defined as the change in position over the change in time over a “long” time. where Dt is a “long” time. vave. Dx Dt =

Average, Instantaneous and Uniform where Dt is a “long” time. “Long” is a relative term, without a specific time period associated with it. It could be hours, days, years – or even seconds. It all depends on what it is being compared to. vavg. Dx Dt =

Average, Instantaneous and Uniform In contrast, we can define an instantaneous velocity as a velocity “at an instant”. Mathematically, we would write this as This means that the instantaneous velocity is just the average velocity as the time interval (Dt) approaches zero, i.e. at an instant. vinst. Dx Dt = limit Dt0

Average, Instantaneous and Uniform To illustrate these concepts, we will tell a little story.

Average, Instantaneous and Uniform A physics student was driving on the interstate when she saw a police car signaling her to pull over.

Average, Instantaneous and Uniform “Do you realize that you were going 70 miles per hour in a 55 mile per hour speed zone?” said the officer.

Average, Instantaneous and Uniform “How can you tell?” asked the student. “I measured your speed with my radar gun,” he replied.

Average, Instantaneous and Uniform “But officer,” insisted the student, “that can’t be. When I left home it was 12:00; now it is 12:15. Also when I left home, my odometer read 99989.9 miles; now it reads 99999.9 miles. In other words, I have traveled 10 miles in 15 minutes.”

Average, Instantaneous and Uniform She continued her argument: “Since speed is distance divided by time, my speed is 10 miles divided by ¼ of an hour.” She took out some paper and wrote: s = d / t = 10 mi / 0.25 h = 40 miles/hour

Average, Instantaneous and Uniform “So Officer, I could not have been going 70 m.p.h. since I was clearly only going 40 m.p.h.” The officer checked her math, found it reasonable and let her go.

Average, Instantaneous and Uniform Question 5: Check with your neighbor – Is this a legitimate story? Why are there two different speeds (70 m.p.h. and 40 m.p.h.)? What does each speed represent?

Average, Instantaneous and Uniform Answer 5: The radar gun (70 m.p.h.) measures instantaneous speed, the speed of the car at a particular instant. The calculated speed (40 m.p.h.) was an average speed, over a “long” time interval – 15 minutes.

Average, Instantaneous and Uniform Question 5 (cont.): How is it possible for the student’s average speed to be 40 m.p.h. and her instantaneous speed to be 70 m.p.h.?

Average, Instantaneous and Uniform Answer 5 (cont.): Her speed changed over time. If her average speed was 40 m.p.h. and her instantaneous speed was 70 m.p.h., then for some time during her trip she was traveling slower than 40 m.p.h.

Average, Instantaneous and Uniform Question 5 (cont.): Should the officer have let her go? If she were in an accident, which speed would matter – her average speed or her instantaneous speed? What does her speedometer measure – her average speed or her instantaneous speed?

Average, Instantaneous and Uniform Answer 5 (cont.): In both cases, the answer is the instantaneous speed. In an accident, the speed at the instant of the accident is what is important. Your speedometer therefore measures your speed at each instant of time. The officer should not have let her go without a ticket.

Average, Instantaneous and Uniform Uniform velocity means that the instantaneous velocity remains constant over time. In your car, you can tell that your speed is constant when the needle on the speedometer stays in the same place.

Average, Instantaneous and Uniform Question 6: Can you think of a device in some cars that allows the driver to maintain uniform speed?

Average, Instantaneous and Uniform Answer 6: Cruise control.

Motion Diagrams Another way to observe motion is through Motion Diagrams. A motion diagram depicts the position of an object at specific instants. Imagine we have a camera taking a picture of the ball below as it moves to the left, but we keep the shutter open. What would the picture look like? (click to animate) - + 1 2 3 4 5 -5 -4 -3 -2 -1 X (meters)

Motion Diagrams The picture looks like a smear because the open shutter takes a picture of the whole motion. - + 1 2 3 4 5 -5 -4 -3 -2 -1 X (meters)

Motion Diagrams Let’s suppose that we kept the shutter open and illuminated the ball with a strobe light for very brief instants. In other words, the ball would only be lit at regular brief instants. What would the picture look like now? (click to animate) - + 1 2 3 4 5 -5 -4 -3 -2 -1 X (meters)

Motion Diagrams These spots represent the position of the ball at particular instants. In this case, every time a new image of the ball is seen, one second has passed, since that is the rate at which the strobe light flashed. We can label the time on the diagram. 5 s 4 s 3 s 2 s 1 s 0 s - + 1 2 3 4 5 -5 -4 -3 -2 -1 X (meters)

Motion Diagrams Question 7: Based on the motion diagram, how would you describe the motion of the ball? Explain your reasoning. 5 s 4 s 3 s 2 s 1 s 0 s - + 1 2 3 4 5 -5 -4 -3 -2 -1 X (meters)

Motion Diagrams Answer 7: The ball is moving with a uniform velocity. You can see this because the ball travels the same displacement in each equal time interval. In this case, Dx = -2 m when Dt = 1 second. Dx = -2 m Dt = 1 s Dx = -2 m Dt = 1 s 5 s 4 s 3 s 2 s 1 s 0 s - + 1 2 3 4 5 -5 -4 -3 -2 -1 X (meters)

Motion Diagrams Answer 7: Since v = Dx/Dt, the value of the velocity is constant at -2 m/s (or 2 m/s left). Dx = -2 m Dt = 1 s Dx = -2 m Dt = 1 s 5 s 4 s 3 s 2 s 1 s 0 s - + 1 2 3 4 5 -5 -4 -3 -2 -1 X (meters)