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9/7/2011 VECTORS – MEASURING FORCE AND MOTION The basics for measuring relationships of forces and objects in physics Scalar Quantity A quantity that can only be described in terms of magnitude Can be added, subtracted, multiplied and divided like ordinaryy numbers Examples include mass, time, temperature, volume, distance, and speed A scalar never mentions directions! Vector Quantity Has both magnitude and direction for a complete description Can be added, subtracted, multiplied, and divided like ordinaryy numbers Examples include force, displacement, velocity, and weight A vector always mentions direction! 1 9/7/2011 Vector types we use in Physics Force – a push or a pull in a given direction Displacement- How far an object moves, and in what direction Velocity – How fast something is moving and in what direction Representation of Vectors Represented graphically by an arrowed line The length of the line g indicates the magnitude of the quantity The direction of the arrow indicates the direction 50 m N 30 m E VECTOR ADDITION Calculating the net effect of two of more vectors acting on the same object. (concurrently) 2 9/7/2011 Techniques for Vector Addition Place the vectors head to tail Draw an arrow from the tail of the first vector to the head of the second vector Th sum off the The th vectors t is i called the Resultant (R) A+B=R Equilibrant: The vector that is equal in magnitude, but in the opposite direction of the resultant B A 3.0 m E 4.0 m E R 7.0 m E 7.0 m W Vector Addition Continued If vectors are heading in the opposite direction the same rules apply, but one of the vectors must be considered to be “negative” A + (-B) = R B 3.0N E A 1.5N W R= 1.5N E Adding Vectors at angles Vectors are not always acting in the same direction, or directly opposing one another! When vectors are at an angle to each other, the pp y laws of math must apply! In physics, we will almost always be concerned with vectors at right angles to each other Therefore, right triangle mathematics will be applied here! 3 9/7/2011 Try this Joe walks 8 meters East then turns and walks 12 meters North. 14 m 56˚ N of E 12 m N 8.0 m E CONSTANT VELOCITY (UNIFORM) MOTION Moving in the same direction at the same speed Some common phrases Motion – the change in position in time Displacement – the distance and direction of an object from a fixed reference point (a vector) Average Velocity – The time rate of change of displacement Instantaneous velocity – velocity of an object at any given point in it’s journey 4 9/7/2011 Uniform Motion A closer look at the terms associated with moving at a constant velocity Distance and direction MUST considered With Constant velocity displacement is always changing by the same amount as time i goes on The graph of displacement vs. time would have a constant slope! The slope is equal to average velocity Displacement (m) Displacement Time (s) rise Δy slope = = run Δx Average Velocity The time rate of change of displacement Can be determined from the slope of a displacement vs. time graph Always Calculated over the entire time period you want to look at! 5 9/7/2011 Equation for Average Velocity v= d t Where - v is average velocity d is displacement t is time Sample Problem A child is pushing a shopping cart at a speed of 1.5 m/s. How long will it take this child to push the cart down an aisle with a length of 9.5 m? v=d/t / Given: v=1.5 m/s d=9.5 m 9.5m t t = 9.5m / 1.5m / s 1.5m / s = t = 6.33 3 s t = 6.3s **This is how every problem should be solved in physics to receive full credit. The given information must be written down clearly. All equations that will be used must be written down completely. And substitutions into the equations must be shown. Displacement vs. time graphs You can tell where the object is relative to a fixed position You can determine the (average) velocity of the object from the slope of the line What is the displacement over each section? What is the average velocity over each section? Over the entire trip? 4 C 3 Displacement (m) D B A 2 E 1 0 -1 0 1 2 3 4 5 6 7 10 Time (s) 6 9/7/2011 An Example for Average Velocity Displacement (m) vs. Time What is the displacement between 0 and 10 sec? 50 What is the displacement between 10 and 35 sec? Displacement (m m) 45 40 35 Total displacement? 30 25 20 What is the average velocity for the first 10 sec? 15 10 5 0 -5 0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 Average for 10 s to 35 sec? Time (s) Another Example Displacement vs. Time 50 A 45 Which object has the greatest average velocity? B 40 Displacement (m m) 35 30 25 What is the displacement of object B? 20 15 10 5 0 0 1 2 3 4 5 6 7 8 Time (s) Another Way to Find Average Velocity Don’t forget the simple way to calculate an average will also work for velocity! Final plus initial velocity divided by 2 is average velocity V= v f + vi 2 7 9/7/2011 Instantaneous Velocity Position (m) Time (s) Velocity of an object at any given point in its journey The slope p of the tangent g line equals velocity at that point! This is the value we read on the speedometer of our cars Practice problems A squirrel descends an 8-m tree at a constant speed in 1.5 min. It remains still at the base of the tree for 2.3 min, and then walks toward an acorn on the ground for .7 min. A loud noise causes the squirrel to scamper back up the tree in 0.1 min to the exact position on the branch from which it started. Which of the following g graphs g p would accuratelyy represent the squirrel’s vertical displacement from the base of the tree? B Time (min) C Time (min) D Position (m) Position (m) Position (m) A Position (m) Time (min) Time (min) Summing Up Displacement considers both magnitude and direction Average velocity can be determined from the slope p vs. time graph g p of a displacement We will focus much more on average velocity than instantaneous velocity 8