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Velocity and Acceleration Group Problem 01 Name_______________________ We are going to begin exploring the motion of objects. To do this we will need a new language, the language of vectors. Vectors are just arrows we use to describe things that naturally have a direction associated with them. Example: Draw an arrow representing an object moving at 10 meters/second to the right. v1 Arrow is 10 units long and points right. v is for velocity. Now the object slows down to 5 m/s, still moving right. Draw this arrow. v2 Arrow is 5 units long and points right. Below are snapshots (taken at equal time intervals) of an object in motion. Can you determine if the object is moving at constant speed, speeding up, or slowing down? It is definitely changing speed but it might be moving right and slowing down or moving left and speeding up. Give an answer assuming the object is moving right. Explain your answer. It is slowing down because the object covers less distance in each time interval. Finish the “velocity diagram” for this motion. Two velocities are given, the object is already moving at the beginning and comes to rest (stops) at the right. Draw arrows with their tails over the object. v1 v2 v3 v4 v5 v6 v7 = 0 1 2 3 4 6 7 5 Now copy all your velocity vectors to the chart below, lining up their tails on the dotted line. Be sure you preserve their length and direction, think “click and drag.” 1 If we say that the object slows down smoothly, then 2 each velocity vector will be shorter than the previous 3 one by the same amount. 4 5 6 7 Velocity and Acceleration Group Problem 01 Name_______________________ Repeat the process assuming the object starts at rest and moves to the left as shown. Draw a velocity diagram and then copy your velocity vectors to the dotted line as before. v7 v6 v5 v4 v3 v 2 v1 = 0 7 6 5 4 2 1 3 1 What is the object doing in this case? Speeding up, 2 slowing down or moving at constant speed? 3 4 The object is speeding up. 5 6 7 If you had to pick one of these velocity vectors to represent the “average velocity,” which one would you pick? Why? I would pick v 4 because it is the middle velocity. Also, average means to add up all the arrows and divide by the number of arrows (including the zero arrow). Another way we will represent velocities (in one dimension) is with a “velocity vs time graph.” We can even show direction if we are careful with minus signs if we are careful. A graph for the first motion (slowing down, moving right) is drawn for you. Can you decide how the graph of the second motion should look? v v 1 2 3 4 5 6 7 t 1 2 3 4 5 6 7 t How can you tell the direction of motion from the graphs? If the graph in above the t-axis the object is moving right and it’s moving left if the graph is below the t-axis. Here we let +/- signs tell us about direction and + = right and - = left. It could have been the other way around, however and we are free to choose any coordinate system (basis) we want. How can you tell speeding up and slowing down from the graphs? If the graph approaches the t-axis as time moves on then the object is slowing down. If the graph becomes farther from the t-axis as time moves on then the object is speeding up. Note there are four possible ways arrange speeding up and slowing down with moving left and moving right. Only two are shown here. Velocity and Acceleration Group Problem 01 Name_______________________ The object’s motion above has changing velocity or acceleration. Since acceleration is complicated we want to break its definition down into steps, a procedure to follow that works in every case. At the heart of this procedure is the change in velocity vector, v . It is important because it allows us to define precisely what we mean by acceleration. For us the acceleration will cover all cases of changing velocity including slowing down as well as speeding up and even turning. v The operational definition of acceleration is: a t While this looks like an equation, it’s better to think of a procedure as follows. 1) Draw the initial and final velocity vectors tail to tail. 2) Draw the change in velocity vector, v1,2 from the tip of the 1st to the tip of the 2nd. (See supplemental page, Change in a Vector) 3) The acceleration vector points in the exact same direction as v . (That’s what the little arrows mean on top of a and v .) 4) The size (length) of the acceleration vector is the size (length) of v divided by the time it takes for the change to occur, t. That is, the time between the initial and final velocities. v1 At right is an example of drawing v . If v1 = 4m/s and v2 v 2 = 12m/s, how long is v ? v1,2 = 8m/s and points right. v1,2 Problem: An object goes from a velocity of 15 m/s right to 6 m/s right in 3 seconds. Use the above operational definition to find the acceleration, its size (magnitude) and its direction, (left or right). v1 v1,2 = 9m/s and points left. Dividing by t (3 seconds) gives the acceleration. v2 a 2 = 3m/s and also points left, the same direction as v1,2 . v1,2 How do the directions of the velocity and acceleration compare? What is the object doing during these 3 seconds? They point in opposite directions and the object is slowing down. How far did the object travel during these three seconds? Hint: What is the average velocity? The average velocity is the average of 6 and 15 or vave = 10.5m/s and points right. The displacement is the average velocity multiplied by the time (3 seconds again) so x = 10.5(3) = 31.5m to the right. What will the objects velocity be in three more seconds if the acceleration stays the same? Hint: Draw a diagram starting with v1 and v . v1 The initial velocity, v1 , is now 6m/s to the right and v1,2 is found from the acceleration v1,2 and time. v1,2 a t 2 = 3(3) = 9m/s and points left, the same direction as a. Lining v2 the v1,2 vector with the tip of v1 allows us to see that v2 = 3m/s and points left.
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