One-Dimensional Motion Introduction to Displacement and Velocity

Objectives Define and calculate displacement Differentiate between displacement and distance Solve velocity problems Differentiate between velocity and speed

Displacement Straight-line distance between the initial and final points Has direction and magnitude Δx, xf-xi X represents position Can be positive or negative Measured in ft, m, km, etc

Distance Not the same as displacement Only has magnitude, no direction Always greater than or equal to displacement

Examples If I walk 2 m east, what is my displacement? If walk 2 m east and 2 m west, what is my displacement? Graph 2 m east The graph should have distance on the y axis, above the x axis is the right, below the x axis is on the left

Vectors Vs Scalars Vectors- Quantities that have magnitude and direction Scalars-quantities that only have magnitude Resultant- vectors added together bbb

Velocity Change of position of an object over an interval of time V=Δx/Δt = xf-xi/t f-t I Has magnitude and direction Vector or scalar? VECTOR!

Example Ms. K takes her dog Zeus for a walk. If they walk for 27 min and travel 1.89 km east, what is their average velocity in meters/sec? 1.2 m/s east

Speed The change in distance over an interval of time Speed=Δx/Δt= xf-xi/t f-t I Only has magnitude no direction! Vector or scalar? They are the same only when moving in a straight line Scalar

Example 1 Polar Bears are extremely good swimmers with an average speed of 2.6 m/s, how far will it have traveled after 2.0 minutes? 240 m

Example 2 Ms. K and Zeus embark on a walk. If they leave Ms. K’s house, travel a distance of 1.2 km and return to the house in 12 minutes and 13 seconds, A) what was their average velocity? B) What their average speed? Give your answer in m/s V=0 Speed- 1.6 m/s The difference between speed and velocity is not just that velocity has a direction!!

Example 3 Zeus and Ms. K embark on a southbound journey. First they walk south at 6.5 km/hr for 1.1 hours. Then they stop to take a nap for 18 minutes and then continue south at 5.5 km/hr for 1.2 hours. A) What was their average velocity? B) What was their displacement? V= 5.3 km/h

Example 4- Honors To qualify for the finals in a racing event a race car must achieve an average speed of 250 km/h on a track with a total length of 2000 m. If a particular car covers the first half of the track at an average speed of 230 km/hr, what minimum average speed must it have in the second half of the event to qualify? V=270 km/hr

One-Dimensional Motion Graphing Position as a function of time

Objectives Draw and interpret distance versus time graphs Interpret what the slope indicates Differentiate between average velocity and instantaneous velocity

Distance Versus Time Graph What is slope? The symbol? What does the slope here indicate????? M=rise/run m= change in y/change of x VELOCITY!! Change in distance (position)/change in time

Example Distance Time What is the slope? What is the velocity? Slope is 0, velocity is 0

Example Distance Time What is the slope? What is the velocity? Slope is -, velocity is -, should be moving to the left

Example Distance Time What is the slope? What is the velocity? Slope is +, velocity is +, moving to the right

Example Distance Time What does this indicate about the velocity of the object at each part? Part a- velocity is +, Part b- velocity is 0, part c-velocity is negative, but is moving faster than at part a because the slope is steeper

Instantaneous Velocity Instantaneous velocity-velocity at a specific point in time Examples? Average Velocity-velocity over a time duration Example? Vf and Vi (instant) v=deltax/deltat (average) Tangent lines are where instantaneous velocities are found

Example Distance 4m 2m 5 10 15 20 time (seconds) 5 10 15 20 time (seconds) What is the velocity between 0-5 seconds? Instant or average?

Example Cont Distance 4m 2m 5 10 15 20 time (seconds) 5 10 15 20 time (seconds) What is the velocity between 5-10 seconds? Instant or average? 2/5= .4m/s

Example Cont Distance 4m 2m 5 10 15 20 time (seconds) 5 10 15 20 time (seconds) What is the velocity at 6 seconds? 9.8 seconds? Instant or average? What is the velocity for 0-20 seconds? What is the slope of the line? 2/5= .4 m/s for both 2/20= .1 m/s,, in this case instant velocity and average velocity will never be the same. Acceleration needs to be constant to have a point in which they are equal

One-Dimensional Motion Acceleration

Objectives Define acceleration Solve acceleration problems Draw and interpret velocity versus time and acceleration versus time graphs

Acceleration The rate that velocity changes, so the change in velocity over the change in time a= Δv/Δt = V f – V i / t f – t I Units- m/s2 Vector or scalar?

Example A sprinter goes from 10 m/s to 15 m/s in 5 seconds, at what rate is the sprinter accelerating? 15-10/5= 1 m/s^2

Example I am driving east at 9.0 m/s and I see a deer and stop in 5.0 seconds A=-1.8 m/s2 east, negative sign indicates that I am slowing down

Instantaneous Versus Average Average acceleration-change in velocity over an interval of time Instantaneous Acceleration-change in velocity at an instant of time

Example A runner starts at a velocity of -1.2 m/s and speeds up constantly during a workout. After 25 minutes the treadmill has a velocity of -6.5 m/s. What is the average acceleration during this time? A= -.00353 m/s2 (3.5 x 10 -3)

Velocity Versus Time Graph Slope=rise over run What does slope indicate here? What about other types of graphs? Acceleration, Acceleration is positive

Example Velocity Time What does this tell us about the acceleration? Acceleration is negative and constant

Example Continued Accel time Acceleration is __________ and is below the time axis because _________________ Constant, the acceleration is negative

Example Continued What does the distance versus time graph look like? A curve starting at the origin that rises steeply at first and then becomes a straight line

Example Velocity Time Draw the acceleration vs time graph and the position vs time Slope changes from positive to negative, direction does not change (remember slope here represent the acceleration). Acceleration graph- should be a straight line above the graph, and a straight line below in the middle Position- should be an s, curve up before the middle, then the curve tapers off

One-Dimensional Motion Uniformly Accelerated Motion

Objectives Solve problems using uniform acceleration equations

Uniformly Accelerated Motion Acceleration is constant What would a velocity versus time graph look like with constant acceleration? Without constant? Equations V f = V i + at Δx = V i Δt + ½ (a t2) Vf2 = Vi2 + 2 a Δx Δx = ½ (Vf – Vi) Δt What variables do we have? 5 variables (vf, vi, Δx, Δt, a) 4 equations If you know 3 variables you can find the other 2 1 happy physic student

Example How long must a runway be for a plane to reach a takeoff velocity of 75 m/s if it accelerates at 20 m/s2? Vi= 0 You can use Vf2 = Vi2 + 2 a Δx Solve for Δx, Δx= 1400 m

Example My tea tumbler falls off my car and slides along 95 South for 75 m. Friction slows my tumbler at 6 m/s2. A)How fast was the car moving when the tumbler fell? B)How long did it take the tumbler to stop? Vi=30 m/s T= 5 s

One-Dimensional Motion Freefall

Objectives Define freefall Solve freefall problems

Freefall If the only force acting upon an object is gravity the object is said to be in freefall No _________________ Considered to be uniform accelerated motion g is the acceleration due to gravity= 9.8 m/s2 When an object is in freefall we will use -9/8 m/s2 Does mass matter? What would a distance versus time graph look like for a ball being thrown in the air?

Example A ball is dropped from a height of 2.0 m. What is the velocity before it strikes the ground? How long did it take to hit the ground? Motion is in the y direction, Δy is negative, when you solve for Vf answer will be positive. The calculator only gives the positive answer! Final velocity should be negative! Vf= -6.3 m/s, time=.64 seconds. You should never get a negative time!

Example Cont Draw the position, velocity, and acceleration graphs Acceleration- constant since it is gravity, horizontal line at -9.8 Velocity-slope is -9.8, so there is a straight line starting at the origin and is below the time axis Distance- line starts at 2, and curves down ward

Example continued How long would it take for the same ball to be thrown up 2m and then fall to the ground? T=.64 x 2= 1.28 m/s (1.3) Since acceleration is constant, and the ending and beginning heights are the same, so the graphs are a mirror image. The time is just doubled.

Example A ball is thrown straight down with a speed of 0.50 m/s from a height of 4.0 m. What is the speed of the ball 0.70 seconds after the ball is released? Vi is negative since the ball is thrown down. Vf= -7.4 m/s (ball is still going down)

Example A 0.25 kg baseball is thrown upward w/ a speed of 30 m/s. Neglect friction. What is the maximum height that the baseball reaches? The final velocity here is 0, that is when the ball reaches the max height. Your initial velocity is + since the ball is traveling upward. Delta y=45.9 m