Lesson 1.2 Position Time Graphs Be able to explain the motion represented in a position time graph Be able to calculate the avg. vel, x, and t for portions of a position time graph. Be able to draw a position time graph given information about avg. vel, x, and t Dec 9 6:23 AM Position (x) Describes the location of an object. Dec 9 6:24 AM 1
Given graph: slope = velocity straight = constant velocity determine position at a time determine disp between two times determine distance Slides needed: Draw a graph given initial position, slope (velocity) and time initial position, final position, and time etc Nov 15 1:42 PM Lesson 1.2 Speed v. Velocity Objectives: 1. Understand the concepts of (instantaneous) speed and (instantaneous) velocity. 2. Understand how to change speed or velocity. Nov 21 3:04 PM 2
Topic 2: Speed v. Velocity Speed (s) is basically how fast an object travels. Velocity (v) is the same as speed, but includes direction. Aug 24 12:17 PM Also, notice that because velocity includes direction, if Al is driving North at 45 mph and Bob is driving South at 45 mph, then they are driving at the same speed but DIFFERENT velocities. So, there are: Aug 24 2:28 PM 3
Two ways to change your speed: Three ways to change your velocity 1. Speed up 1. Speed up 2. Slow down 2. Slow down 3. Turn MOVE ME! MOVE ME! Aug 24 2:29 PM Note.when we talk about speed or velocity, we are talking about instantaneous speed and instantaneous velocity because we are referring to the object s speed or velocity at one exact moment in time. Nov 14 2:34 PM 4
Topic 3: Acceleration What is acceleration? Okay so far, in all of the problems we ve done, our objects have been moving at a consta velocity. As a result, the object s velocity (which was constant) was the same as the object average velocity. But what do we do if the object s velocity (instantaneous velocity) changes? Well, I ll tell you.we find the object s acceleration. So, new definition: Acceleration (a): The rate at which an object s velocity changes. So, let s think about this. First of all, this definition refers to the velocity changing. Usually, when you think of an object accelerating, you think about it speeding up, right? Like, you might say, Hey, Carl, people will be mad if you get on the freeway at 25 mph, so you should accelerate before you get on the freeway. So here you re telling Carl to speed up right? However, technically, to accelerate all an object needs to do is change its velocity.so remember, there are: Sep 1 2:46 PM Two ways to Three ways to change your speed: change your velocity 1. Speed up 1. Speed up 2. Slow down 2. Slow down 3. Turn Note.when we talk about speed or velocity, we are talking about instantaneous speed and instantaneous velocity because we are referring to the object s speed or velocity at one exact moment in time. MOVE ME! MOVE ME! Aug 24 2:29 PM 5
SO an object is accelerating any time it changes its velocity any time it does any one of these three things. Remember the only time you need to worry about spilling your coffee in a car, is if you re accelerating, right? This happens if you speed up (the coffee sloshes back), slow down (the coffee sloshes forward) or turn (the coffee sloshes out the side). Now, acceleration has a numeric value it measures the rate at which an object s velocity changes, or how quickly the object s velocity changes. Here are a couple examples: Sep 12 10:43 AM Example 1: So, imagine you are in a large school bus at a red light. You look out the window and see a Corvette, or some crazy fast sports car next to the bus. The light turns green, and both the bus and the Corvette speed up to a speed of 45 miles per hour. Picture this! In your head! The Corvette will get to a speed of 45 mph very quickly, but the bus will take a while to get to this speed. Here, the Corvette has a large acceleration, because its velocity changed (from 0 to 45 mph) very quickly but the bus has a small acceleration because its velocity changed (from 0 to 45 mph) very slowly. Sep 12 10:44 AM 6
Example 2: Similarly, imagine two identical cars driving at 50 miles per hour. The driver of Car A stops his car by slowly pressing the brake pedal. The driver of Car B stops his car by driving into a brick wall. Both cars are changing velocity they are both going from 50 mph to 0 mph. But we say Car B has a larger acceleration because its velocity changes much more quickly. Sep 12 10:44 AM Test your knowledge: Answer these questions! 1) Can an object change its velocity without changing its speed? If so, how? If not, why? 2) Can an object change its speed without changing its velocity? If so, how? If not, why? Sep 12 10:45 AM 7
3) Look at the situations below. If the object in the example is accelerating, write accelerating next to it. If it is not accelerating, write constant velocity next to it. a) A car speeds up from a stop light. Accelerating Constant Velocity b) A biker hits the brakes to avoid hitting a cat. Speeding Up Slowing Down Turning c) A car drives straight North with the cruise control set at 60 mph. d) You are walking at 2.0 m/s when you turn a corner without speeding up or slowing down. e) A flower pot falls off a windowsill. f) A skateboarder rolls down a steep hill. Sep 12 10:46 AM 4) Which object will have a bigger acceleration? a) Ball A or Ball B: b) A motorcycle speeding up, or a semi truck speeding up. c) An empty shopping cart which you push as hard as you can, or a full shopping cart which you push as hard as you can. Sep 12 10:47 AM 8
How to calculate acceleration Cool, so now we have a pretty good idea of what acceleration is. But how do we calculate it? Well: Here, the variables represent: a = acceleration v i = initial velocity = the velocity the object begins with v f = final velocity = the velocity the object ends up with t = time = the amount of time it takes for the object to get from its initial velocity to its final velocity. So, let s do a quick example. For the first example, I m going to use mph (miles per hour) for the velocity, just because I think it s a little more familiar to you. Then, for the next example, I ll use m/s (meters per second). Sep 12 10:50 AM Example 3: Many car magazines will tell you a car s Time to go from zero to 60. This is just a measure of how long it will take the car to speed up, from a dead stop, to a speed of 60 mph. So, suppose a fancy sports car can go from 0 to 60 mph in 5 seconds. Let s find its acceleration. Given: v i = 0 mph(because the car isn t moving at the start) v f = 60 mph (because that is the speed the car reaches) t = 5 sec(because that is how long it takes to speed up) Find:a =? Equation:a = v f v i t Solve:a = 60 mph 0 mph = 60 mph = 12 mph 5 sec 5 sec sec Okay. Great. We found the acceleration. It is 12 mph/sec. Awesome. Well, what the h*** does this tell us? Well, here s what this means: a = 12 mph/sec means that every second, the velocity increases by 12 mph. Think about the speedometer as you speed up in a car. As you speed, up the needle points to increasing speeds, right? Look at the speedometers shown below. These indicate speed in miles per hour. (mph). Every second, the speed (velocity) increases by 12 mph. Sep 12 10:53 AM 9
t = 0 s t = 1 s t = 2 s t = 3 s t = 4 s t = 5 s Sep 12 10:54 AM Another way we could think about this is by using a chart. Look, the velocity began at zero, and then increased by 12 mph each second. Check it out: SO, in this example, the car s acceleration is 12 mph / sec because every second, the velocity increases by 12 mph Sep 12 11:07 AM 10
Test your knowledge: Try this on your own: 1) A large train is moving at 48 miles per hour when the conductor sees a cow on the track. He pulls the brake, and after 8 seconds, the train has stopped, just before hitting the cow. PHEW! a) Calculate the train s acceleration using the equation on the previous page. Be sure to write all givens and show all work. b) So, the train s velocity s by mph every, c) Notice.what does it mean that the acceleration here is negative? Sep 12 11:07 AM Sep 12 11:08 AM 11
Sep 12 11:10 AM Units of acceleration. Okay so now, hopefully, you have a good understanding of acceleration. The only problem is we can t measure acceleratio in units of mph / sec because mph (miles per hour) isn t metric. So, we re going to, from now on, measure our velocity in meters per second (m/s). This means our acceleration will be measured in meters pers second per second (m/s / s), or more simply, meters per second squared (m/s 2 ). Let s do an example: Example 4: A car is driving at 23 m/s when the driver hits the brakes. After 4 seconds, the car is only going 11 m/s. Find the car s acceleration. Given: v i = 23 m/s (because that s how fast the car is going when hits he the brakes) v f = 11 m/s (because that is the speed the car slows down to) t = 4 sec(because that is how long it takes to slow down) Find:a =? Equation:a = v f v i t Solve:a = 11 m/s 23 m/s = 12 m/s = 3 m/s = 3 m/s 2 4 sec 4 sec s Okay. So, this means: Every second, the car s velocity decreases by 3 m/s. Sep 12 11:10 AM 12
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Example 6: Roy is driving his car at 8 m/s when he hits the gas pedal for 5 seconds, causing the car to accelerate at 6 m/s 2. What velocity does the car reach? Sep 12 11:13 AM Before we learn about freefall, we need to learn what a force is. Force: (F) A push or pull on an object. Examples: Force of Roy, Force of Gravity, Force of Friction, Force of Electricity, Force of Air Resistance, etc Anyway, we ll talk more about forces next chapter. So if there is a big force on an object, that means something is pushing or pulling it hard. A force tries to change the motion of an object. Please note though that how much force is on an object has NOTHING to do with how fast it is moving. Sep 12 2:18 PM 14
Freefall: An object is in freefall if the only force acting on it is gravity. Examples: A ball which has just been dropped off a building A ball which has just been thrown DOWN off a building A ball which has just been thrown UP into the air. In all of these examples, the only force on the ball is gravity. That s why they all move down faster and faster. Sep 21 3:39 PM Galileo discovered that: ALL objects in freefall accelerate at the acceleration of gravity (a g ). On earth, a g = 9.8 m/s 2 although for simplicity, we ll use a g = 10 m/s 2 Notice, a g is different on different planets, etc.: On the moon, a g 1.6 m/s 2 On Jupiter, a g 100 m/s 2 Sep 21 3:41 PM 15
Keep in mind that in freefall there is NO air resistance. In the real world, there IS air resistance, but we ll usually ignore it because unless the object is either very light or moving very quickly, the air resistance doesn t have a significant effect on the object s motion. Additionally, it is very difficult to calculate the effects of air resistance. Anyway, if an object is in freefall, Every second the object s velocity decreases by 10 m/s Sep 21 3:42 PM Sep 21 3:44 PM 16
Sep 21 3:51 PM So, we ve learned how to calculate the displacement of an object which is traveling at a constant velocity. But what if the object s velocity is changing? To answer this, let s look at two simple examples: Example 1: Suppose a car travels at 30 m/s for 6 seconds. How far will the car travel in this time? Note: This example is (hopefully) simple to solve because the car is moving at a constant velocity so the car travels 30 meters every second. But what if the car s velocity is changing? Sep 22 10:59 AM 17
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Sep 22 11:05 AM Example 3: So, let s use both options to solve this: A motorcycle driving at 24 m/s slows down at 3 m/s 2 for 5 seconds. How far will it travel in this time? Sep 26 11:06 AM 19
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So, in general, my suggestion is: Use Option 1 if you know both v f and v i Use Option 2 if you know a. Sep 26 9:34 AM Now, let s take a closer look at the first chart. Notice that because the ball speed is changing, it doesn t travel the same distance every second. Sep 22 11:06 AM 21
Sep 26 12:41 PM if vi = 0 Notice that even though 60 m is twice as far as 30 m, it takes LESS than twice as much time. This is because when the ball rolls 60 m, it takes less time to roll the last 30 m because it is going faster by that time. Sep 22 11:09 AM 22
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