# chap. 3: kinematics in two or three dimensions: vectors hw3: chap. 2: pb. 51, pb. 63, pb. 67; chap...

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• Chap. 3: Kinematics in Two or Three Dimensions: VectorsHW3: Chap. 2: Pb. 51, Pb. 63, Pb. 67; Chap 3:Pb.3,Pb.5, Pb.10, Pb.38, Pb.46Due Wednesday, Sept. 16

• Variable Acceleration; Integral CalculusDeriving the kinematic equations through integration:For constant acceleration,

• Variable Acceleration; Integral CalculusThen:For constant acceleration,

• Displacement from a graph of constant vx(t)Solve for displacementDisplacement is the area between the vx(t) curve and the time axis+x-xSIGN0

• Displacement from graphs of v(t)What to do with a squiggly vx(t)?

make t so small that vx(t) does not change much

t1t2Displacement is the area under vx(t) curvet vxtVelocity does not need to be constant

• Graphical Analysis and Numerical IntegrationSimilarly, the velocity may be written as the area under the a-t curve.However, if the velocity or acceleration is not integrable, or is known only graphically, numerical integration may be used instead.

• Review QuestionA ball is thrown straight up into the air. Ignore air resistance. While the ball is in the air the accelerationA) increasesB) is zeroC) remains constantD) decreases on the way up and increases on the way downE) changes direction

Examples:DisplacementVelocityaccelerationDistancespeedtime

Examples:

• 2D VectorsMagnitude and direction are both required for a vector!How do I get to Washington from New York?Oh, its just 233 miles away.

• Vector Addition: GraphicalWhen we add vectorsWe add vectors by drawing them tip to tail ABThe resultant starts at the beginning of the first vectorand ends at the end of the second vector

• Vector Addition Question1)2)3)Which graph shows the correct placement of vectors for +

• Vector Addition Question1)2)3)Which graph shows the correct resultant for +

• Vector Subtraction: GraphicalWhen you subtract vectors, you add the vectors opposite. - = + -

• Addition of VectorsGraphical MethodsThe parallelogram method may also be used; here again the vectors must be tail-to-tip.

• Multiplication of a Vector by a ScalarA vector can be multiplied by a scalar c; the result is a vector c that has the same direction but a magnitude cV. If c is negative, the resultant vector points in the opposite direction.

• Vector Addition: ComponentsIf the components are perpendicular, they can be found using trigonometric functions.

• Vector Addition: ComponentsWe dont always carry around a ruler and a protractor, and our result isnt always very precise even when we do. In this course we will use components to add vectors.However, you should still always draw the vector addition to help you visualize the situation.What are components here?xyAxAyParts of the vector that lie on the coordinate axes

• Vector Addition: ComponentsOnce we have the components of C, Cx and Cy, we can find the magnitude and direction of C.CxCymagnitudedirection

• Unit VectorsUnit vectors have magnitude 1. Using unit vectors, any vector can be written in terms of its components:

• Adding Vectors by ComponentsExample 3-2: Mail carriers displacement.A rural mail carrier leaves the post office and drives 22.0 km in a northerly direction. She then drives in a direction 60.0 south of east for 47.0 km. What is her displacement from the post office?

• Vector KinematicsIn two or three dimensions, the displacement is a vector:

• Vector KinematicsAs t and r become smaller and smaller, the average velocity approaches the instantaneous velocity.

• Vector KinematicsThe instantaneous acceleration is in the direction of = 2 1, and is given by:

• Vector KinematicsUsing unit vectors,

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