Basics of graphing motion And studying their slopes S.Caesar

In the study of kinematics, we consider a moving object as a particle. Motion in 1 Dimension v  In the study of kinematics, we consider a moving object as a particle. A particle is a point-like mass having infinitesimal size and a finite mass. Motion in One Dimension (2053)

= 6 m - 2 m Motion in 1 Dimension Displacement The displacement of a particle is defined as its change in position. x Dx = x - xo = 6 m - 2 m = 4 m (m) -6 -4 -2 2 4 6 Note: Motion to the right is positive Motion in One Dimension (2053)

= -6 m - 6 m Motion in 1 Dimension Displacement The displacement of a particle is defined as its change in position. x Dx = x - xo = -6 m - 6 m = -12 m (m) -6 -4 -2 2 4 6 Note: Motion to the left is negative Motion in One Dimension (2053)

= (2 m) - (-6 m) Motion in 1 Dimension Displacement The displacement of a particle is defined as its change in position. x Dx = x - xo = (2 m) - (-6 m) = 8 m (m) -6 -4 -2 2 4 6 Note: Motion to the right is positive Motion in One Dimension (2053)

Graphs tell a story about motion We will use 3 different graphs to tell our story: Position (distance) vs time graph (p-t graph) Velocity vs time graph (v-t graph) Acceleration vs time graph (a-t graph)

The importance of the SLOPE The slope of a line on a position-time graph gives you the velocity p Pick any two points on the line. Since slope = rise run In this case the rise = ∆position the run = ∆time Slope = rise = (6m – 2m) = 4m = .5 m/s run = (12s – 4s) 8s 6m 2m 4s 12s t

Velocity is represented displacement-time graph Motion in 1 Dimension Average velocity The average velocity of a particle is defined as x x1 x2 t1 t2 Dx Dt Velocity is represented by the slope on a displacement-time graph t Motion in One Dimension (2053)

Figure 2-24 Graph of position vs Figure 2-24 Graph of position vs. time for an object moving at a uniform velocity of 11 m/s.

Act out these “stories” Position time

Position time

Position time

Position time

Find the velocities? Position Position 20 30 10 10 time time

The average speed of a particle is defined as Motion in 1 Dimension Average speed The average speed of a particle is defined as Motion in One Dimension (2053)

Velocity In everyday language we use “speed” and “velocity” interchangeably. In physics they are different. speed = distance (total length of traveled path) time (total time of travel) velocity = displacement (distance if you would have gone directly there)

Displacement The displacement is written: Left: Displacement is positive. Right: Displacement is negative.

= 6 m - 2 m Motion in 1 Dimension Displacement The displacement of a particle is defined as its change in position. x Dx = x - xo = 6 m - 2 m = 4 m (m) -6 -4 -2 2 4 6 Note: Motion to the right is positive Motion in One Dimension (2053)

Vector vs Scalar If someone asks you for directions they are actually asking you for TWO pieces of information…. They want to know: HOW FAR WHAT DIRECTION They are asking you for a vector A vector has magnitude and direction

Vector How do you get to CVS Pharmacy? You go 1.5 miles SOUTH direction Magnitude

Vector direction The direction of a vector can be indicated in many ways…. Directions of the compass (ex: South, 600 E of N) Up, down, left, right “+”, “-” The direction of “+” has to be indicated So “ – 10m” would be 10m in the negative direction

There are very few straight lines in real life…… What do we do with curves on a position-time graph? Look at the following graph and try to figure out what is happening……(teacher slides)

Acceleration is represented Motion in 1 Dimension Average acceleration The average acceleration of a particle is defined as the change in velocity Dvx divided by the time interval Dt during which that change occurred. v v1 v2 t1 t2 Dv Dt Acceleration is represented by the slope on a velocity-time graph t Motion in One Dimension (2053)

Displacement, velocity and acceleration graphs x Motion in 1 Dimension Displacement, velocity and acceleration graphs x The slope of a displacement-time graph represents velocity t v The slope of a velocity-time graph represents acceleration t a t Motion in One Dimension (2053)

Displacement, velocity and acceleration graphs x Motion in 1 Dimension Displacement, velocity and acceleration graphs x The area under a velocity-time graph represents displacement. Dx t v The area under an acceleration-time graph represents change in velocity. Dv t a Dt t Motion in One Dimension (2053)

Definitions of velocity and acceleration Motion in 1 Dimension Definitions of velocity and acceleration Average velocity Average acceleration Motion in One Dimension (2053)

? Motion in 1 Dimension For constant acceleration An object moving with an initial velocity vo undergoes a constant acceleration a for a time t. Find the final velocity. vo ? a time = 0 time = t Solution: Eq 1 Motion in One Dimension (2053)

What are we calculating? t a DV Motion in One Dimension (2053)

Instantaneous Velocity This means how fast are you going THIS INSTANT IN TIME? How do you find out this information while you are driving? Look at your speedometer AND your compass. How do you find out this information from a graph? Find the tangent of the curve AT THAT INSTANT.

How do you find the tangent on a curve?

What does the tangent tell you? It gives you the slope of the line AT THAT POINT. The slope of the line on the p-t graph gives you the magnitude of the velocity The direction of the slope (+ or -) gives you the direction of the velocity (For those of you in calculus – you can find the “derivative” of the curve at that point to get the tangent)

tA

Acceleration Acceleration

Name 3 things in your car that can cause acceleration Gas pedal (sometimes called the “accelerator”!) Brake Steering wheel (What???!!) Any change in direction is also a change in velocity (since it can change direction)

So acceleration is any kind of change in velocity. Going faster Going slower Changing direction

What does NON-constant acceleration look like? Don’t panic – we won’t calculate anything with changing acceleration!!

“Constant” acceleration In real life How long does it take your car to go from 0-40 mph ? How long WOULD it take for your car to go from 60-100mph? (that’s ALSO a change of 40 mph in case you weren’t paying attention) Don’t EVEN think about trying it! Both the above situation had same acceleration

So what do we mean by “Constant Acceleration” For the sake of simpler math (yeah!) we will assume that the acceleration of something happens at the same rate no matter what. That would mean that it would take the same amount of time for your car to accelerate from 60-100 mph as it did for it to accelerate from 0-40 mph.

∆ v = v2 – v1 ∆ v = +10 m/s ∆ v = 0 m/s ∆ v = -10 m/s ∆ v = -10 m/s before after before after A ∆ v = +10 m/s D ∆ v = 0 m/s +10 m/s +20 m/s +20 m/s +20 m/s E B ∆ v = -10 m/s ∆ v = -10 m/s +30 m/s +20 m/s +10 m/s C F ∆ v = -20 m/s ∆ v = -10 m/s +10 m/s -20 m/s -10 m/s -10 m/s Most Negative _____ _____ _____ _____ _____ _____ Most Positive C B E F D A

If these vectors (lovingly called arrows If these vectors (lovingly called arrows!) represent motion, what is happening? Notice that the acceleration vectors are the SAME length. That means the acceleration is CONSTANT! a V1 a V2 V3 So we know the object is speeding up, going in the same direction How do we represent HOW MUCH it sped up? By using acceleration vectors

- + a V1 a V2 V3 Velocity vectors are positive. Acceleration vectors are positive.

- + a V1 V2 a V3 We know the object is speeding up, in the negative direction. Velocity vectors are negative. Acceleration vectors are negative.

+ - V1 V2 a a V3 We know the object is slowing down, in the negative direction. Velocity vectors are negative. Acceleration vectors are positive.

Acceleration There is a difference between “negative acceleration” and “deceleration”: “Negative acceleration” is acceleration in the negative direction as defined by the coordinate system. “Deceleration” occurs when the acceleration is opposite in direction to the velocity.

+ + + - - + Scenario Sign of velocity Sign of acceleration The driver has stepped on the accelerator and the car is just starting to move forward. The car is moving forward. The brakes have been applied. The car is slowing down, but has not yet come to rest. The car is moving backward. The brakes have been applied. The car is slowing down, but has not yet come to rest. + + + - - +

So remember….. When the velocity vectors and the acceleration vectors are in the SAME direction, the object is speeding up. When the velocity vectors and the acceleration vectors are in the OPPOSITE direction, the object is slowing down.

The important question…… How fast does that change in velocity take place?

Acceleration Acceleration is the rate of change of velocity. How fast will the car be going after 1 sec? How fast will the car be going after 2 seconds? How fast will the car be going after 5 seconds?

What will be the velocity after 5 seconds? Acceleration Acceleration is a vector, although in one-dimensional motion we only need the sign. The previous image shows positive acceleration; here is negative acceleration: What will be the velocity after 5 seconds?

Our new equation…. a = ∆v = v2 – v1 = the change in velocity ∆t ∆t the time it took to change We abbreviate it as a = ∆v t The SI unit of velocity is m/s(meter/second) The SI Unit of time is s (seconds) ∆ -delta means change

Back to graphing…… Interactive MINI Demo Things to remember….. “Cart on a ramp” Discussion…… Things to remember….. Accelerating motion will be a curve on a p-t graph Since we are having a constant acceleration, the a-t graph will always be horizontal In a v-t graph –constant acceleration has a steady slope (straight line)

Motion in 1 Dimension More Graphs More Graphs Practice Session Consider the situations below and plot graph More Graphs Practice Session Motion in One Dimension (2053)

-6 -5 -4 -3 -2 -1 1 1 2 3 4 5 6 Motion in One Dimension (2053)

-6 -5 -4 -3 -2 -1 1 1 2 3 4 5 6 Motion in One Dimension (2053)

-6 -5 -4 -3 -2 -1 1 1 2 3 4 5 6 Motion in One Dimension (2053)

6 5 4 3 2 1 1 2 4 6 8 10 12 -1 -2 -3 -4 -5 -6 Motion in One Dimension (2053)

m 6 5 4 3 2 1 1 2 4 6 8 10 12 -1 s -2 -3 -4 -5 -6 Motion in One Dimension (2053)

6 m 4 2 2 4 6 8 10 12 -2 s -4 -6 2 v 1 t 4 8 12 (s) -1 (m/s) -2 -3 Motion in One Dimension (2053)

6 m 4 2 2 4 6 8 10 12 -2 s -4 -6 2 v +8 m 1 +4 m t 4 8 12 (s) -1 (m/s) -12 m -2 -3 Motion in One Dimension (2053)

4 8 12 16 20 24 28 (s) x 4 8 12 16 20 24 28 (m) 1 2 3 t 5 Motion in One Dimension (2053)

(s) x 4 8 12 16 20 24 28 (m) 1 2 3 t 5 1 2 3 4 5 t (s) 6 8 10 v (m/s) Displacement 25 m Motion in One Dimension (2053)

(a) Velocity vs. time and (b) displacement vs (a) Velocity vs. time and (b) displacement vs. time for an object with variable velocity. You make the v-t graph for the p-t graph shown to the right.

V1 = 0 (released from rest), origin is to the left, positive is to the right, and no friction Final location p t v t a t

Study more of these type of graphs Examples of using area under the curve of the v-t graph to determine the displacement. Study more of these type of graphs

Constant velocity v 10 m/s * 4 s = 40 m 10 m/s 10 m/s * 6 s = 60 m 2 s If an object travels at 10 m/s for 4 seconds, how far did it travel? 10 m/s * 4 s = 40 m 10 m/s If an object travels at 10 m/s for 6 seconds, how far did it travel? 10 m/s * 6 s = 60 m 2 s 4 s 6 s t

Motion with acceleration v If an object starts from rest and accelerates to 10 m/s in 6 seconds, how far did it travel? 10 m/s d = ½ (10m/s)(6 s) d = 30 m 2 s 4 s 6 s t

Motion with acceleration and initial velocity ≠ 0 10 m/s If an object starts from an initial velocity of 5 m/s and accelerates to 10 m/s in 6 seconds, how far did it travel? 5 m/s t 2 s 4 s 6 s d = area of rectangle + area of triangle d = (5 m/s)(6 s) + ½ (10 m/s – 5 m/s)(6 s) d = 30 m + 15 m = 45 m

Calculate the total stopping distance Calculate the total stopping distance. Reaction time for this driver is 0.50s.

Figure 2-16 Example 2–9. Graph of v vs. t.

What is the displacement of the object from 4s to 6s? v 10 m/s If an object starts from an initial velocity of 5 m/s and accelerates to 10 m/s in 6 seconds, how far did it travel between seconds 4 and 6? 7.5 m/s 5 m/s t 2 s 4 s 6 s d = area of triangle + area of rectangle d = ½ (10 - 7.5 m/s)(2 s) + (7.5 m/s)(2 s) d = 2.5 m + 15 m = 17.5 m

A special case of constant acceleration……. GRAVITY!!

Demo……

How much gravity is in the tube? Think about it……..

Name 3 ways you can get a piece of paper to fall as fast as a book. Mini demonstrations are done the students in groups Note the student answers-brainstorm session

If you drop an object in the absence of air resistance, it accelerates downward at 9.8 m/s2. If instead, you THROW it downward, its downward acceleration AFTER release is: 1. less than 9.8 m/s2 2. 9.8 m/s2 3. more than 9.8 m/s2

You are throwing a ball straight up in the air You are throwing a ball straight up in the air. At the highest point, the ball’s …. 1. velocity and acceleration are zero. 2. velocity is nonzero but its acceleration is zero. 3. acceleration is nonzero, but its velocity is zero. 4. velocity and acceleration are both nonzero.

Air Resistance (a) A ball and a light piece of paper are dropped at the same time. (b) Repeated, with the paper wadded up.

Study the direction of the velocity and acceleration as the object is thrown up. Study the direction of the velocity and acceleration as the object is falling down. An object thrown into the air leaves the thrower's hand at A, reaches its maximum height at B, and returns to the original position at C. Examples 2-12, 2–13, 2–14, and 2–15.

Figure 2-30 Problem 11.

Figure 2-40 Problem 73.

Problem Solving Skills 1. Read the problem carefully 2. Sketch the problem 3. Visualize the physical situation 4. Strategize 5. Identify appropriate equations 6. Solve the equations 7. Check your answers Motion in One Dimension (2053)