Ch 2 Velocity ~Motion in One Dimension~

Scalar versus Vector Scalar – quantity that only has magnitude –In the previous slide, which is the scalar? Vector – quantity that has magnitude and direction. –In the previous slide, which is the vector? What other examples of scalars can you think of? Vectors?

Three friends drive their four wheelers a distance of m. Will they end up in the same place? Why or why not. No, m is a scalar and does not specify a specific direction. Now suppose they drive m due East. Will they end up in the same place? Why or why not. Yes, m East is a vector. It has direction Scalar versus Vector

Reference Frames A reference frame is an established coordinate system used to define motion. The “zero” mark in a reference frame is called the reference point. Reference Frame Reference Point

Position An object’s position pertains to its location in the reference frame (on the number line). –For example the position of the truck. –The position of the bus. Position values can be positive, negative, or zero. The variables used for position are “x 1 ” and “x 2 ”. Common units for position are meters (m). Reference Frame X 35 X -5m 3m3m

Distance Distance is the measure of separation between two objects. It is given the variable “d” and is measured in meters (m). If the positions are known, distance is calculated as follows. The distance between the buoys be? The distance from the shore to each buoy is. Distance is a scalar quantity m 3m and 16m 16m 3m3m

Displacement When an object is displaced, it is moved from an initial position (d 1 ) to a final position (x 2 ). Displacement (variable  x) is a measure of the change in position of an object after it has moved.

Example Displacement - Distance John travels east along a straight highway and passes mile marker 260. John continues until mile marker 150 and then doubles back to mile marker 175. What is Johns displacement from marker 260? +85 miles What is John’s distance from marker 260? 85 miles What is the total distance John traveled? 135 miles

Example Displacement - Distance John travels west along a straight highway and passes mile marker 260. John continues until mile marker 150 and then doubles back to mile marker 175. What is Johns displacement from marker 260? -85 miles What is John’s distance from marker 260? 85 miles What is the total distance John traveled? 135 miles

Speed Speed: The measure of how fast something is moving. It is the Rate at which distance is covered Speed = distance / time 235

Average Velocity Average Velocity – the change in position of an object over a given time interval. Finial positionInitial position Initial time Final time Vector or Scalar??

Instantaneous Speed You raced your 4-wheeler over a 30 mile long track in 1.5 hours. What is your average speed? 20mph Watch the animation. Is the speed always 20mph?

Instantaneous Speed The actual speed at any “instant” in time. –e.g. The speedometer of a car is the instantaneous speed. Average speed is total distance / total time.

Speed versus Velocity Speed has no direction! –e.g. 25 miles per hour (mph) Velocity has magnitude and direction! –e.g. 25 mph west. Which is the vector and which is the scalar?

Time (s) Position (m) Position vs. Time Graph

What is the position of the object at t = 10s? What is the position of the object at t = 40s? What is the position of the object at t = 60s? Position vs. Time Graph Position (m) Time (s)

Position vs. Time Graph Position (m) Time (s)

Position vs. Time Graph Position (m) Time (s) + Slope + Velocity Moving forwards - Slope - Velocity Moving backwards 0 Slope 0 Velocity Not moving

Position vs. Time Graph Position (m) Time (s) During what time period(s) is the object moving forward? During what time period(s) is the object moving backwards? During what time period(s) is the object not moving?

Position vs. Time Graph Position (m) Time (s) Find the velocity between 0s-15s Find the velocity between 15s-30s Find the velocity between 30s-50s

Average Acceleration The rate at which the Velocity changes. –Scalar or vector? Average acceleration Change in velocity Change in time What are the units of acceleration?

Practice Problem – Avg Acceleration The velocity of a car increases from 2.0 m/s at 1.0 s to 16 m/s at 4.5 s. What is the car’s average acceleration? We know: We want:

Average Acceleration Can an object speed up and have a negative average acceleration? –YES! Let’s see an example how!

Practice Problem – Avg Acceleration Simon rolls backwards faster and faster down his driveway. He starts at -2.0 m/s and is moving at -9.0 m/s 2.0 s later. What is his average acceleration? Negative a! AHH H!!!!

Average Acceleration Can an object slow down and have a positive average acceleration? –YES! Let’s see an example how!

Practice Problem – Avg Acceleration Simon rolls up the other side of the ramp and slows down from-9.0m/s to - 2m/s in 2s. What is his average acceleration? AHH H!!!!

Instantaneous Velocity Draw a tangent line at the point that corresponds to that instant in time Find the slope of that tangent line at 5.0s, 2.0s, and 9.0s. Rise - Δd Run - Δt

Position Time Graphs Velocity Time Graphs Velocity vs. Time Graph Time (s) Velocity (m/s) Position vs. Time Graph Position (m) Time (s) Position–Time graphs lets us calculate velocity. Velocity–Time graphs lets us calculate displacement (  x).

Area Calculations In order to calculate area, you will need to know how to find the area of different shapes. TrapezoidTriangleRectangle/Square

Velocity vs. Time Graph Time (s) Velocity (m/s) Displacement equals the area between the curve and the x-axis. (  d = Area) Find the displacement between 10s-25s.

“Negative” Area Area above the x- axis indicates positive displacement. Area below the x- axis indicates negative displacement. Negative velocity means negative displacement Velocity vs. Time Graph Time (s) Velocity (m/s)

Velocity vs. Time Graph Time (s) Velocity (m/s) Displacement equals the area under the curve (  d = Area) Find the distance traveled between 40-55s

Velocity vs. Time Graph Time (s) Velocity (m/s) Displacement equals the area under the curve (  d = Area) Find the distance traveled between 25-65s

Velocity vs. Time Graph Time (s) Velocity (m/s) Displacement equals the area under the curve (  d = Area) Find the displacement traveled between 10-55s

Velocity vs. Time Graph Time (s) Velocity (m/s) Find the displacement between 40-75s 337.5m left

Velocity-Time ΔvΔv ΔtΔt What is the average acceleration of the object over the first 2 s? Slope of velocity-time graph is the average acceleration!!

Velocity-Time Graphs During what time period(s) does the object have a positive acceleration? During what time period(s) does the object have a negative acceleration? During what time period(s) is the object not accelerating?

Velocity-Time Graphs During what time period(s) is the sign of the velocity and acceleration opposite?

Instantaneous Acceleration – acceleration at one instant in time What is the instantaneous acceleration at 4 s? Draw the tangent line to that point. Find the slope of that line. ΔtΔt ΔvΔv

Constant Acceleration Acceleration that does not change in time is uniform or constant acceleration. On a Velocity-Time Graph, constant acceleration is a straight line

Velocity – Time Graph The slope is the Acceleration

Velocity of an Object with Constant Acceleration Constant Acceleration = Uniform Acceleration What would the graph of a versus t look like?

Velocity of an Object with Constant Acceleration D vs t v vs t a vs t

Zero Acceleration, Constant Speed We can use small triangles to visualize the distance traveled per increment of time. The same amount of distance is covered in the same amount of time. The speed of the object remained constant.

Negative Acceleration We can use small triangles to visualize the distance traveled per increment of time. The second triangle is much smaller that the first triangle. The larger size means that the body traveled more distance in the same increment of time The object was moving slower during the second increment. The object experienced a negative acceleration.

Positive Acceleration We can use small triangles to visualize the distance traveled per increment of time. The second triangle is much larger that the first triangle. The larger size means that the body traveled more distance in the same increment of time The object was moving faster during the second increment. The object experienced a positive acceleration.

Graphs of Motion Label the three distance v. time graphs below as either accelerating positive, accelerating negative, or zero acceleration.

Velocity versus Time Label the three velocity versus time graphs below as either – accelerating positive –accelerating negative –zero acceleration. Simply read the values directly from the graph.

Displacement when Acceleration and Time are Known We will combine 2 equations we know to get another equation. Substitute v 2 into first equation

t v1v1 v2v2

Displacement when Velocity and Acceleration are Known We’ll use this equation when time is not known. Solve for t Substitute

Summary of Displacement Under Uniform Acceleration Missing dMissing aMissing v 2 Missing t

Why is the distance traveled the same for both 6s and 9s? Example 4b #3 Distance traveled for the 1 st 6s. Distance traveled for the 1 st 9s.

Example Displacement - Distance John travels east along a straight highway and passes mile marker 260. John continues until mile marker 150 and then doubles back to mile marker 175. What is Johns displacement from marker 260? +85 miles What is John’s distance from marker 260? 85 miles What is the total distance John traveled? 135 miles

Example Average Velocity At 1:00 PM a car, traveling 94 km/h west, is 17 km west of your school. Where will it be at 3:30 PM? The car traveled 235km, where is it at 3:30pm? 17km west + 235km west 252km west of the school