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Unformatted text preview: Kinematics in 2D; Vectors
Chapter Outline GENERAL PHYSICS Vectors and Scalars PHY 302K Addition and Subtraction of Vectors
Chapter 3: Kinematics in 2D; Vectors Components of a Vector
Displacement, Velocity, and Acceleration in 2D Maxim Tsoi Motion in 2D; Projectile Motion
Relative Velocity Physics Department,
The University of Texas at Austin
http://www.ph.utexas.edu/~tsoi/302K.htm
302K  Ch.3 302K  Ch.3 Vector and Scalar Quantities Some Properties of Vectors Magnitude and Direction Equality of two vectors • Two vectors A and B are equal if they have the same magnitude • A scalar quantity is completely specified and point in the same direction (A=B, AB) by a single value with an appropriate unit
and has no direction (e.g., temperature,
volume, mass, speed, time intervals)
• A vector quantity is completely
specified by a number and appropriate units plus a direction (e.g., velocity, A, A , A, A displacement) 302K  Ch.3 302K  Ch.3 Some Properties of Vectors Some Properties of Vectors Laws of vector addition • The rules for adding vectors are conveniently described by graphical methods
• To add vector B to vector A: Adding more than two vectors • The rules for adding vectors are conveniently described by graphical methods
• To add four (N) vectors A, B, C, and D:
• Draw 2nd (3rd, etc.) vector with its tail • Draw B with its tail starting from the starting from the tip of the 1st (2nd, etc) tip of A • R=A+B+C+D is the vector drawn from • The vector R=A+B is the vector the tail of the first vector to the tip of drawn from the tail of A to the tip of B the last vector Total displacement is the vector sum
of the individual displacements 302K  Ch.3 Total displacement is the vector sum
of the individual displacements 302K  Ch.3 1 Some Properties of Vectors Some Properties of Vectors Commutative law of addition Associative law of addition • The rules for adding vectors are conveniently described by graphical methods
• When two vectors are added, the sum is • The rules for adding vectors are conveniently described by graphical methods
• When three or more vectors are added, their independent of the order of the addition: sum is independent of the way in which the
individual vectors are grouped together: • A+B=B+A • A+(B+C)=(A+B)+C Total displacement is the vector sum
of the individual displacements 302K  Ch.3 • All vectors must
have the same units
and type of quantity Total displacement is the vector sum
of the individual displacements 302K  Ch.3 Some Properties of Vectors Some Properties of Vectors Negative of a vector Subtracting vectors • The rules for adding vectors are conveniently described by graphical methods
• The negative of the vector A is defined as the • The rules for adding vectors are conveniently described by graphical methods
• Make use of the definition of the negative of a vector that when added to A gives zero for the vector vector sum: • We define subtraction AB as vector –B added • A+(A)=0 to vector A:
• A – B = A + (B) • The vectors A and A have the same
magnitude but point in opposite directions A
A 302K  Ch.3 302K  Ch.3 Some Properties of Vectors Components of a Vector Multiplying a vector by a scalar Adding vectors using their projections • The rules for adding vectors are conveniently described by graphical methods • The projections of vector A along coordinate axes
are called the components Ax and Ay of the vector • m – positive the product mA is a vector that has the • Any vector can be completely described by its same direction as A and magnitude mA components • m – negative the product mA is directed opposite A Ax A cos A
5A B Ay A sin ½ B 2
2
A Ax Ay Ay Ax tan 1 A=Ax+Ay the signs of the components
depend on the angle a vector A can be specified either
with Ax and Ay or with A and ` 302K  Ch.3 302K  Ch.3 2 Coordinate Systems Components of a Vector
Using components to add vectors Description of a location in space Rx Ax Bx • Cartesian coordinate system (rectangular coordinates) (x, y) R y Ay B y • Polar coordinate system (r, ) x r cos y r sin tan y
x r x2 y2 2
2
R Rx R y Example 3.2 302K  Ch.3 Components of a Vector Ax Bx 2 Ay B y 2 tan Ry
Rx Ay B y
Ax B x 302K  Ch.3 Position, Velocity, and Acceleration Rotation of a coordinate system Position and displacement vectors • Motion of a particle is completely known if its position is • In many applications/problems it is convenient to express the components known as a function of time of a vector A in a coordinate system having axes that are not horizontal
and vertical but are still perpendicular to each other • Position of a particle is described by its position vector r • In different coordinate systems the components of the same vector must
be modified accordingly r is drawn from the y’ origin of a coordinate x’ system to the particle Ax A cos Ax ' A Ay A sin Ay ' 0 • Displacement vector is the difference between its final
position vector and its initial position vector: r r f ri
302K  Ch.3 302K  Ch.3 Position, Velocity, and Acceleration Position, Velocity, and Acceleration Average and instantaneous velocity vectors • Average velocity of a particle during time
interval t is the displacement of the particle
divided by the time interval: r
v
t Average and instantaneous acceleration vectors • Average acceleration of a particle
is the change in its instantaneous
velocity vector divided by the time
interval: • Instantaneous velocity is the limit of the
average velocity as t approaches zero: r
v lim
t 0 t v f v i v a
t f ti
t • Instantaneous acceleration is the
limit of the average acceleration as
t approaches zero: v
a lim
t 0 t • The magnitude of the instantaneous velocity Various changes can occur when a particle
accelerates: magnitude and/or direction! vector v=v is called speed 302K  Ch.3 302K  Ch.3 3 2D Motion with Constant Acceleration Projectile Motion Kinematic equations in components v f vi a t e. g., a baseball in motion Assumptions v xf v xi a x t v yf v yi a y t r f ri v i t 1 a t 2
2 • The freefall acceleration g is constant over the
range of motion and is directed downward 2
1 x f xi v xi t 2 a x t 2 y f yi v yi t 1 a y t
2 • The effect of air resistance is negligible • 2D motion at constant acceleration is equivalent to two independent
motions – in x and y directions – having constant accelerations ax and ay Example 3.5 302K  Ch.3 The path of a projectile, called trajectory, is always a parabola Example 3.6 302K  Ch.3 Projectile Motion Projectile Motion Trajectory of a projectile Trajectory of a projectile • Vector expression for the position vector of the projectile: • Reference frame: y is vertical and positive is upward ay=g, ax=0 r f ri v i t 1 g t 2
2 • At t=0 the projectile leaves the origin (xi=yi=0) with velocity vi
which makes an angle i with x xi 0
v xi v i cos i yi 0
v yi v i sin i 2
1 x f xi v xi t 2 a x t 2 y f yi v yi t 1 a y t
2 x f v xi t v i cos i t
y f v yi t 1 a y t 2 v i sin i t 1 gt 2
2
2
• Projectile motion is superposition of two motions: 2 g
y tan i x 2 2v cos 2 x
i i constantvelocity motion in the horizontal direction y=axbx2 parabola freefall motion in the vertical direction 302K  Ch.3 302K  Ch.3 Projectile Motion Projectile Motion Horizontal range and maximum height of a projectile Horizontal range and maximum height of a projectile
• Various trajectories of a projectile launched with the same speed at different angles • Two points of the projectile trajectory
are especially interesting to analyze: The peak point (R/2, h) The landing point (R, 0)
• R – horizontal range
• h – maximum height v yf v yi a y t 0 v i sin i gt A h v yi t 1 a y t 2 v i sin i 2 tA v i sin i 1 v i sin i 2 g g
g R v xi t B v i cos i 2t A v i cos i 302K  Ch.3 v i sin i
g 2 v 2 sin 2 i i 2g 2v i sin i v i2 sin 2 i g
g R
Example 3.7 v i2 sin 2 i
g RMAX v i2
g 302K  Ch.3 4 Relative Velocity and Relative Acceleration Relative Velocity and Relative Acceleration Observations made in different frames of reference • Observers in different frames of reference may measure different Observations made in different frames of reference • Observers in different frames of reference may measure different positions, velocities, and accelerations for a given particle positions, velocities, and accelerations for a given particle • Two observers moving relative to each other generally • Two observers moving relative to each other generally do not agree on the outcome of a measurement 302K  Ch.3 do not agree on the outcome of a measurement 302K  Ch.3 Relative Velocity and Relative Acceleration SUMMARY Galilean transformation equations • A particle located at point (A) • SCALAR quantities have only a numerical value; no direction • Two reference frames S and S’ • VECTOR quantities have both magnitude (>0) and direction; obey • S’ moves to the right relative the laws of vector addition to S with constant v0 • R=A+B is the vector drawn from the tail of A to the tip of B • at time t: r ' r v0 t r' r v0
t
t v' v t
t Vectors • The x (y) component Ax (Ay) of the vector A is equal to the
projection of A along the x (y) axis of a coordinate system v' v v 0 • Components of a vector A:
• The acceleration of a particle measured Ax A cos Ay A sin by an observer in one frame of reference
is the same as that measured by any other observer moving with constant a' a velocity relative to the first frame 302K  Ch.3 302K  Ch.3 SUMMARY
Kinematics in 2D
• If a particle moves with constant acceleration a and velocity vi and position ri
at t=0, its velocity and position vectors at some later time t are: v f vi a t r f ri v i t 1 a t 2
2 • For 2D motion in xy plane component expressions of these two vector
expressions describe two independent 1D motions in x and y directions
• Projectile motion type of 2D motion under constant acceleration where ax=0
and ay=g
• Velocity of a particle in different frames of reference moving with constant
velocity relative to each other are related by: v ' v v0 302K  Ch.3 5 ...
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