Ch. 2 Section 2 and Ch. 3: Acceleration Day 1 Ch. 2 Section 2 and Ch. 3: Acceleration

Motion in this class so Far… In one direction only! Focus on how objects move, not why Distance, Displacement, Speed, Velocity Position vs. Time Graphs

This Week’s Objectives: Students will: Be able to explain what acceleration is and describe how it affects motion. Continue to analyze and interpret motion graphs. Interpret information given in word problems and solve for unknown quantities.

Acceleration – Ch. 2 Pg. 48

Calculating Avg. Acceleration

Calculating Avg. Acceleration

Solution

Practice In Class Assignment: Weekly Homework Assignment: Pg. 49: Practice B Problems Weekly Homework Assignment: Pg. 69-70 #’s: 12 - 22

We will look at examples on the following slides. Acceleration Day 2 Note that Acceleration has direction and magnitude (Thus it is a VECTOR.) The sign of the acceleration vector can inform us if the object is speeding up or slowing down. But we must also know whether the object is moving with a + or – velocity to really know what is happening. We will look at examples on the following slides.

Constant Positive Velocity

Constant Negative Velocity

Positive Velocity and Positive Acceleration

Positive Velocity and Negative Acceleration

Negative Velocity and Negative Acceleration

Negative Velocity and Positive Acceleration

Summary

Motion with Velocity vs. Time Graphs

Motion with Velocity vs. Time Graphs

So…

If this graph is positive acceleration, What do you suspect the slope of a Velocity versus Time graph tells us?

That’s Right! Slope = Average Acceleration

#1 Check Your Understanding Describe the motion depicted by the following velocity-time graphs. In your descriptions, make reference to the direction of motion (+ or - direction), the velocity and acceleration and any changes in speed (speeding up or slowing down) during the various time intervals (e.g., intervals A, B, and C). When finished, click the button to see the answers.

#1 The object moves in the + direction at a constant speed - zero acceleration (interval A). The object then continues in the + direction while slowing down with a negative acceleration (interval B). Finally, the object moves at a constant speed in the + direction, slower than before (interval C).

#2

#2 The object moves in the + direction while slowing down; this involves a negative acceleration (interval A). It then remains at rest (interval B). The object then moves in the - direction while speeding up; this also involves a negative acceleration (interval C).

#3

#3 The object moves in the + direction with a constant velocity and zero acceleration (interval A). The object then slows down while moving in the + direction (i.e., it has a negative acceleration) until it finally reaches a 0 velocity (stops) (interval B). Then the object moves in the - direction while speeding up; this corresponds to a - acceleration (interval C).

Determining the Area on a v-t Graph In a Velocity vs. Time Graph, the AREA UNDER THE CURVE is equal to the total displacement. The shaded area is representative of the displacement during from 0 seconds to 6 seconds. This area takes on the shape of a rectangle & can be calculated using the appropriate equation: base x height So the total Displacement = 30m/s x 6 s = 180 m

The shaded area is representative of the displacement during from 0 seconds to 4 seconds. This area takes on the shape of a triangle can be calculated using the appropriate equation: ½ Base x Height

How can you calculate the area under this curve? The shaded area is representative of the displacement during from 2 seconds to 5 seconds. This area takes on the shape of a trapezoid. How can you calculate the area under this curve?

Acceleration Recall Equation: a = Dv /Dt Day 3 Acceleration Recall Equation: a = Dv /Dt “acceleration equals change in velocity divided by the time interval” But what if we wanted to know the Displacement an object has traveled while accelerating? This question leads us to the end of Ch. 2, and the introduction of the Kinematic Equations.

Observation Time! Watch the following video clip of the best basketball player to ever play the game. (Other than Mr. Pacton of course…) Answer the following questions in your notebook: Is this athlete’s velocity constant at all times? When is the velocity changing the most? When is the velocity changing the least? http://www.youtube.com/watch?v=wqPRdzrjWpU&feature=related @ 2:20

Today’s Goals: Kinematics Equations Add the Kinematics Equations to your formula page and (more importantly) know what they can be used for. Know what each variable stands for in the kinematics equations. Understand how these equations are derived. Use these equations in sample problems.

Kinematics Equations The goal of Kinematics is to accurately and completely describe the motion of real world objects. On the following slides we will derive each equation.

Kinematics Equations These equations can be used only when velocity is constant (acceleration = 0 m/s/s) or acceleration is constant. They can never be used for any time period during which the acceleration is changing!!!! Kinematics (Greek κινειν,kiein) meaning to move

1.) This equation comes from the idea of ‘averages’. If you average the initial and final velocity over a period of time Dt, and then multiply by the time spent traveling, you would find the total displacement.

2.) This equation comes directly from the definition of acceleration. http://id.mind.net/~zona/mstm/physics/mechanics/kinematics/EquationsForAcceleratedMotion/Origins/Velocity/Origin.htm

3.) This equation relates displacement, initial velocity, acceleration, and time. As you can see, this equation depends on time. It is often referred to as the TIME DEPENDENT equation. http://id.mind.net/~zona/mstm/physics/mechanics/kinematics/EquationsForAcceleratedMotion/Origins/Displacement/Origin.htm

4.) This equation relates displacement, velocity, and acceleration. As you can see, this equation does not depend on time. It is often referred to as the TIME INDEPENDENT equation. We derive this equation by combining the other two kinematics equations in this section, and, through substitutions, eliminating the time variable. http://id.mind.net/~zona/mstm/physics/mechanics/kinematics/EquationsForAcceleratedMotion/Origins/TimeIndependent/Origin.htm

Kinematics Equations Xi df Vi vf a t Initial Position Final Position Variables used in these equations: Xi Initial Position df Final Position Vi Initial Velocity, the velocity at the start of the acceleration vf Final Velocity, the velocity at the end of the acceleration. a Average Acceleration (When the acceleration is changing the kinematics equations cannot be used. Therefore, the average acceleration, which is a constant, is used.) t Time, this is the time period of the acceleration.

Problem-Solving Strategy Figure out what the problem is telling/ asking you. Constructing a diagram of the physical situation might help. Identify and list the given variables. Identify and list the unknown variables. Identify and list the equation which will be used to determine unknown information from known information. Substitute known values into the equation and use appropriate algebraic steps to solve for the unknown information. Check your answer to insure that it is reasonable and mathematically correct.

Check Your Understanding An airplane starts from rest and then accelerates down a runway at 4 m/s2 for 30.0 s until it finally lifts off the ground. Determine the distance traveled (aka displacement) before takeoff. Formula? x = Vit + ½at2 Given? a = 4 m/s2 t = 30.0 s Vi = 0 m/s x = ? Solution Formula: x = Vit + ½at2 = (0) (30.0 s) + ½ ( 4 m/s2 ) (30.0 s )2 = 1800 m

Given? Formula? x = Vit + ½at2 Solution Formula x = Vit + ½at2 A car starts from rest and accelerates for 5.0 seconds over a distance of 125 m. Determine the acceleration of the car assuming it is constant. Given? starts from rest Vi = 0 t = 5.0 seconds x = 125 m a - ? Formula? x = Vit + ½at2 Solution Formula x = Vit + ½at2 125 m = (0 )(5) + ½ (a) (5)2 250/25 = 10 a = 10 m/s2

Formula(s)? Vf = Vi + at And x =(vi + vf) t 2 Or A race car accelerates from 18 m/s to 48.0 m/s in 3 seconds. Assuming constant acceleration, determine the acceleration of the car and the distance it traveled. Formula(s)? Vf = Vi + at And x =(vi + vf) t 2 Or x = vit + ½at2 Given? Vi = 18 m/s Vf = 48.0 m/s t = 3 seconds a = ? Solution Formula Vf = Vi + at 48 = 18 + (a) (3) 30 = 3 a a = 10 m/s2 and x = ((18 + 48)/2)x3 = 99 m Note: either of these two formula’s will work to find d!

A bike accelerates from rest to a speed of 5 A bike accelerates from rest to a speed of 5.0 m/s over a distance of 50.0 m. Determine the acceleration of the bike assuming it is constant. Given? from rest Vi = 0 m/s Vi = 5.0 m/s d = 50.0 m a = ? Formula? Vf2 = Vi2 + 2a(Dx) Solution Formula Vf2 = Vi2 + 2aDx (5)2 = (0)2 + 2 (a) (50) 25 = 100 a a = 0.25 m/s2

An engineer is designing the runway for an airport An engineer is designing the runway for an airport. Of the planes which will use the airport, the lowest acceleration rate is 3 m/s2. The takeoff speed for this plane is 60 m/s. What is the minimum allowed length for the runway in order for this plane to take off? Given? Vi = o m/s Vf = 60 m/s a = 3 m/s2 Dx= ? Formula? Vf2 = Vi2 + 2aDx Solution Formula Vf2 = Vi2 + 2aDx (60)2 = (0)2 + 2 (3)Dx 3600 = 6Dx Dx = 600 m

Solution: Formula Vf2 = Vi2 + 2aDx (0)2 = (100)2 + 2 (-4)(Dx) Can a plane land safely at an airport that has its longest runway of 1000 m if its landing speed is 100 m/s and its breaks are capable of slowing it down by 4 m/s every second? Given? Vi = 100 m/s a = - 4 m/s2 Vf = 0 m/s Dx = ? Needs to be less than 1000 for plane to land Formula? Vf2 = Vi2 + 2aDx Solution: Formula Vf2 = Vi2 + 2aDx (0)2 = (100)2 + 2 (-4)(Dx) 10000 = - 8Dx -Dx = 1250 m The plane cannot land

It was once recorded that a car left skid marks which were 400 m in length. Assuming that the car skidded to a stop with a constant acceleration of -2.00 m/s2, determine the speed of the car before it began to skid. Given? Dx = 400 m a = - 2.00 m/s2 Vf = 0 m/s Vi = ? Solution Formula Vf2 = Vi2 + 2aDx (0)2 = Vi2 + 2 ( -2) (400) Vi2 = 1600 Vi = 40 m/s Formula? Vf2 = Vi2 + 2aDx

Quiz A dragster accelerates to a speed of 120 m/s over a distance of 400 m. Determine the acceleration (assume constant) of the dragster.

Do objects speed up as they fall? Write down as many observations as possible as you view the following video clip of an object falling. (Minimum 3) Question #1: Do objects speed up as they fall? Falling Object 1

Yes! Objects accelerate as they fall. The force of gravity (to be discussed next quarter) pulls on falling objects, causing them to accelerate downward, toward the center of the Earth. “The acceleration due to gravity” is represented by the letter g g = 9.8 m/s2

Watch the next video clip Watch the next video clip. Observe how two different objects dropped at the same time fall. Answer the following question: Question #2: Do objects of different masses accelerate at the same rate? How do you know this from the video? Falling Object s 2

Because: 1. Constant Acceleration 2. Motion in 1 direction Question #3: Can the kinematics equations be applied to falling objects? YES! Because: 1. Constant Acceleration 2. Motion in 1 direction Required in order for the Kinematics Equations to apply.

Dy =1/2(vi + vf)t 2. Vf = Vi - gt 3. Dy = Vit - ½gt2 Kinematics Equations for Objects in Freefall Dy =1/2(vi + vf)t 2. Vf = Vi - gt 3. Dy = Vit - ½gt2 4. Vf2 = Vi2 - 2gDy Are they different???? No, not really

Summary

OPEN YOUR BOOKS! Pg. 63: Example Problem F