Units of Chapter 2 Reference Frames and Displacement (2-1) Average Velocity (2-2) Instantaneous Velocity (2-3) Acceleration (2-4) Motion at Constant Acceleration (2-5) Falling Objects (2-7) Problem Solving!!! (2-6…and all chapter!)

Mechanics MotionEnergy KinematicsDynamics Ch.2 Introduction Kinematics – Describes how objects move Dynamics – Force, and why objects move the way they do.

2-1 Reference Frames and Displacement Any measurement of position, distance, or speed must be made with respect to a frame of reference…this problem arises because there are many different ways to measure things!

2-1 Reference Frames and Displacement Question: How fast is the person on the train walking if… You are on the train with him? You are standing on a sidewalk watching the train go by? Train is moving west at 80 km/h

2-1 Reference Frames and Displacement You will get two different answers just based off of where you are standing! If, with respect to the train, he is moving at 5 km/h, then he is moving at _____ km/h _________ if you measured his speed from the sidewalk! Train is moving west at 80 km/h

2-1 Reference Frames and Displacement Another way to state this is to say that, with respect (or in reference) to the Earth, the passenger is moving at 85 km/h to the west. Train is moving west at 80 km/h

2-1 Reference Frames and Displacement Now, what happens if the passenger instead begins walking to the back of the train (towards the east) at 5 km/h? Assume that you are still watching from the sidewalk… Train is moving west at 80 km/h

2-1 Reference Frames and Displacement Try it out! Going on vacation, Liz is on a plane bound for FL traveling due south at 7.50*10 2 km/h. If she leaves her seat and begins heading north at 3.5 km/hr, what is her speed with respect to the ground?

2-1 Reference Frames and Displacement For 1-D motion, we will most often be using the x-axis as our reference frame. (Horizontal motion) The y-axis will be used for vertical motion, such as throwing an object straight up or for free fall.

2-1 Reference Frames and Displacement Distance vs. Displacement Distance refers to how far something has traveled (works like your car odometer). → Speed Displacement is how far you are from your original starting point. → Velocity

2-1 Reference Frames and Displacement Distance vs. Displacement To calculate distance, you merely sum how far your “travel path” took you. To calculate displacement, use this formula:

2-1 Reference Frames and Displacement Displacement The symbol Δ (delta) means change. Formally, it states “final value – initial value”. In the formula, “x” refers to position! x 2 : final position x 1 : initial position

2-1 Reference Frames and Displacement The displacement is written as:

2-1 Reference Frames and Displacement The displacement is written as: Displacement is positive.Displacement is negative.

2-2 Average Velocity Speed: how far an object travels in a given time interval Velocity includes directional information:

2-2 Average Velocity For the diagram given below, calculate the object’s average speed and average velocity. Assume that the entire trip took 12.0 s.

2-2 Average Velocity The position of a runner as a function of time is plotted as moving along the x-axis of a coordinate system. During a 3.00 s time interval, the runner’s position changes from x 1 = 50.0 m to x 2 = 30.5 m. What is the runner’s average velocity? His average speed?

2-2 Average Velocity How far can a cyclist travel in 2.5 h along a straight road if her average velocity is 18 km/h?

2-3 Instantaneous Velocity The instantaneous velocity is the average velocity in the limit as the time interval becomes infinitesimally (VERY) short. These graphs show (a) constant velocity and (b) varying velocity. (2-3)

Frames of Reference and Velocity Mr. Scott is traveling to school at a rate of 35 m/s. Sam is in a hurry, and is driving behind Mr. Scott at 45 m/s. If Mr. Scott looks in his rearview mirror and sees Sam approaching, how fast does it appear that Sam is moving? If Mr. Scott first notices Sam when he is 200 m away, how long will it take Sam’s car to reach his vehicle?

2-4 Acceleration Acceleration is the rate of change of velocity. When we say that an object is accelerating, we mean that it is either speeding up or slowing down. What is velocity?

2-4 Acceleration A car accelerates along a straight road from rest to 75 km/h in 5.0 s. What is the magnitude of the average acceleration?

2-4 Acceleration Acceleration is a vector, therefore it has magnitude and direction. However, in 1-D motion, the only a + or – sign is needed to indicate direction, as long as the coordinate system being used is defined. What is the direction of this car’s velocity? Its acceleration?

2-4 Acceleration CHECK YOUR UNDERSTANDING If the velocity of an object is zero, does that mean that its acceleration must be zero? If an object’s acceleration is zero, does that mean that its velocity is zero?

2-4 Acceleration An automobile is moving to the right along a straight highway, which we choose to be the positive x-axis. Then the driver puts on the brakes. If the initial velocity (when the driver hits the brakes) is 15.0 m/s, and it takes the car 5.0 m/s to slow down to 5.0 m/s, what was the car’s average acceleration?

2-4 Acceleration Deceleration is used specifically when an object is slowing down. When you are decelerating, does your acceleration have to be negative??? Deceleration occurs when the acceleration is opposite in direction to the object’s velocity.

2-4 Acceleration The instantaneous acceleration is the average acceleration, in the limit as the time interval becomes infinitesimally short. (2-5)

The average velocity of an object during a time interval t is The acceleration, assumed constant, is 2-5 Motion at Constant Acceleration

In addition, as the velocity is increasing at a constant rate, we know that Combining equations (2-7) through (2-9), we find that: (2-8) (2-9)

2-5 Motion at Constant Acceleration We can also combine these equations so as to eliminate t: We now have all the equations we need to solve constant-acceleration problems. (2-10) (2-11a) (2-11b) (2-11c) (2-11d) The kinematic equations

2-5 Motion at Constant Acceleration A person riding a bike accelerates from 2.5 m/s to 7.0 m/s in 10.0 s while travelling down a hill. What was the biker’s acceleration during this time? How far did the biker travel?

2-5 Motion at Constant Acceleration You are designing an airport for small planes. One kind of plane that might use the airfield must before takeoff reach a speed of at least 27.8 m/s, and can accelerate at a rate of 2.00 m/s 2. (a) If the runway is 150m long, can this airplane reach the required speed for take off?

2-6 Solving Problems 1. Read the whole problem and make sure you understand it. Then re-read the problem, and underline the problem’s key points/info. 2. Decide what the object(s) under study are and what the time interval is in the problem. 3. Draw a diagram and choose coordinate axes. 4. Write down the known (given) quantities, and then the unknown ones that you need to find. 5. What physics applies here? Plan an approach to a solution.

2-6 Solving Problems 6. Decide which of the equations relate the known and unknown quantities? Are the equations valid for the situation in the problem? You will want to use an equation where you are only solving for one unknown. Solve algebraically for the unknown variable before plugging in the given numbers, and check that your result is sensible. 7. Calculate the solution and round it to the appropriate number of significant figures. Check to make sure that you have the correct units for the unknown you solved for. 8. Look at the result – is it reasonable? Does it agree with a rough estimate? 9. Check the units of your answer again (if solving for time, for example, you want to make sure your answer has a unit of time attached!).

2-7 Falling Objects Near the surface of the Earth, all objects experience approximately the same acceleration due to gravity. This is one of the most common examples of motion with constant acceleration.

2-7 Falling Objects In the absence of air resistance, all objects fall with the same acceleration, although this may be hard to tell by testing in an environment where there is air resistance. Acceleration due to Earth’s gravity (in the direction of the Earth): g = 9.80 m/s 2

2-7 Falling Objects The acceleration due to gravity at the Earth’s surface is approximately 9.80 m/s 2.

2-7 Falling Objects Suppose that a ball is dropped (meaning v o = 0) from a tower that is 70.0m high. How far will the ball have fallen after time t 1 = 1.00 s, and t 2 = 2.00 s?

2-7 Falling Objects Suppose now that the ball is thrown straight down from the 70.0 m tower with a velocity of 3.00 m/s instead of being dropped. What would its speed be after 1.00 s and 2.00 s? How does this compare with our previous calculations?

2-7 Falling Objects A person throws a ball upward into the air with an initial velocity of 15.0 m/s. Calculate (a) how high the ball goes and (b) how long the ball is in the air before it comes back to his hand.

2-7 Falling Objects

What is the acceleration of the ball at the very top of its flight path in the previous example? Why? During the ball’s upward motion, the ball’s acceleration and velocity are in the ____________ direction. During the ball’s downward motion, the ball’s acceleration and velocity are in the ____________ direction.

Summary of Chapter 2 Kinematics is the description of how objects move with respect to a defined reference frame. Displacement is the change in position of an object. Average speed is the distance traveled divided by the time it took; average velocity is the displacement divided by the time. Instantaneous velocity is the limit as the time becomes infinitesimally short.

Summary of Chapter 2 Average acceleration is the change in velocity divided by the time. Instantaneous acceleration is the limit as the time interval becomes infinitesimally small. The equations of motion for constant acceleration are given in the text; there are four, each one of which requires a different set of quantities. Objects falling (or having been projected) near the surface of the Earth experience a gravitational acceleration of 9.80 m/s 2.