Kinematics- Motion in Two Dimensions. Math Review - Trigonometry. Trigonometry. Trigonometry. Trigonometry. Pythagorean theorem:. Trigonometry Example. FIND THE VALUE OF h o. Trigonometry. Vectors and Scalars.
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Trigonometry Pythagorean theorem:
Trigonometry Example FIND THE VALUE OF ho
Vectors and Scalars Vector - Quantity deals inherently with both magnitude (size) and direction Scalar -Quantity that has no associated direction; specified completely by a number and units • Scalars: • Temperature • Distance • Speed • Time • Mass • Energy • Vectors: • Displacement • Velocity • Acceleration • Momentum • Force
Vectors Simplified • For a complete explanation of vectors, check out this website: Vectors Explained
Mass versus Weight • Mass • Scalar quantity (no direction) • Measures the amount of matter in an object • Weight • Vector (points toward center of Earth) • Force of gravity on an object On the moon, your mass would be the same, but the magnitude of your weight would be less.
Scalars and Vectors • Vectors: • Represented by arrows • Direction of the arrow gives the direction of • the vector • By convention, the length of a vector arrow is proportional to the magnitude of the vector 8 lb 4 lb
The length of the arrow represents the magnitude (how far, how fast, how strong, etc, depending on the type of vector). The arrow points in the direction of the force, motion, displacement, etc. It is often specified by an angle. Vectors Vectors are represented with arrows 5 m/s 42°
Components of Motion • Focus of the previous chapter One Dimensional Motion • Object moving in a straight line was considered to be moving along one of the x or y axes • What if the motion is not along an axis? • Can we still apply the kinematic equations? Yes
Two Dimensional Motion HOWEVER, both the x and y coordinates are needed to describe the diagonal motion of the ball =Two Dimensional Motion Motion of the ball moving in a straight line along the y axis = one dimensional motion V Vy y (Vy) Vx Motion of the ball moving in a straight line along the x axis = one dimensional motion x (Vx)
Two Dimensional Motion • Consider the motion of a ball moving diagonally • ** Think of the ball moving in the x and y directions simultaneously • Meaning • It has a velocity in the x direction (Vx) AND • It has a velocity in the y direction (Vy) • ** The combined velocity components describe the actual motion of the ball
Components of Motion • Ball has a constant velocity (V) in a direction at an angle theta relative to the x axis • Velocities in the x and y directions are obtained by resolving the velocity vector into components of motion in these directions • Magnitude of the Vx and Vy components NOTICE
To Add Vectors Using the Method of Components • Utilize trigonometric functions • Pythagorean Theorem:
The Components of a Vector Example A displacement vector has a magnitude of 175 m and points at an angle of 50.0 degrees relative to the x axis. Find the x and y components of this vector.
Kinematic Equations Two dimensional motion utilizes the same kinematic equations previously described
Equations of Kinematics in Two Dimensions The x part of the motion occurs exactly as it would if the y part did not occur at all, and vice versa.
Equations of Kinematics in Two Dimensions Example:A Moving Spacecraft In the x direction, the spacecraft has an initial velocity component of +22 m/s and an acceleration of +24 m/s2. In the y direction, the analogous quantities are +14 m/s and an acceleration of +12 m/s2. Find (a) x and vx, (b) y and vy, and (c) the final velocity of the spacecraft at time 7.0 s.
Equations of Kinematics in Two Dimensions Example:A Moving Spacecraft In the x direction, the spacecraft has an initial velocity component of +22 m/s and an acceleration of +24 m/s2. Find (a) x and vx
Equations of Kinematics in Two Dimensions Example:A Moving Spacecraft In the y direction, the initial velocity component is +14 m/s and the acceleration is +12 m/s2. Find y and vy
Equations of Kinematics in Two Dimensions Find (c) the final velocity of the spacecraft at time 7.0 s.
Equations of Kinematics in Two Dimensions Can go one step further…
Vector Addition & Subtraction Geometric (Graphical) Methods • Triangle Method • Parallelogram Method • Polygon Method ** Need to be drawn to scale Use a protractor to measure theta Analytical (computational) Method “COMPONENT METHOD”
Vector Addition and Subtraction • Vectors: • Quantities with both magnitude and direction have to be added in a special way • Simple arithmetic can be utilized to add vectors that are in the same direction • But, simple arithmetic cannot be used if the two vectors are not along the same line
Vector Addition and Subtraction TRIANGLE METHOD 3 m 5 m 8 m
Vector Addition and Subtraction Sam walks 6 meters East and 2 meters North – Find the resultant displacement vector (red arrow) (Hint: Vector sum of A and B) 2.00 m 6.00 m
Vector Addition and Subtraction R 2.00 m 6.00 m
Vector Addition and Subtraction 6.32 m 2.00 m 6.00 m
General Rules for Graphically Adding Two Vectors • Tail-to-Tip (Triangle method)Method of Adding Vectors • Draw 1 of the vectors to scale • Draw the 2nd vector to scale placing its tail to the tip of the 1st vector make sure direction is correct • Arrow drawn from the tail of the 1st vector to the tip of the 2nd vector represents the sum, or resultant, of the 2 vectors
General Rules for Graphically Adding Two Vectors • Parallelogram Method • Two vectors are drawn starting from a common origin • Two vectors are constructed as the adjacent sides • Resultant Diagonal drawn from the common origin Vy V Vx
“Component Method” • Procedure for Adding Vectors by the Component Method • Resolve vectors to be added into their x and y components. (Use the acute angles (those <90 degrees) between the vectors and the x axis, and indicate the directions of the components by plus or minus signs) • Add all the x components and all the y components together vectorially to obtain the x and y components of the resultant, or vector sum (This is done algebraically by using plus and minus signs)
Procedure continued… • Express the resultant vector by using… - Pythagorean theorem: - Finding the angle of direction – relative to the x axis- by taking the inverse tangent (tan-1) of the absolute value of the ration of the y and x components
NOTES • Wilson book – Fig 3.10 b and c (pg 75) • Wilson book – Fig 3.11 a and b (pg 76)
Relative Velocity • Motion is relative to some reference frame