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Dec 28, 2015
Chapter 4 2D and 3D MotionDefinitions
Uniform circular motion
Motion in Two DimensionsUsing + or signs is not always sufficient to fully describe motion in more than one dimension Vectors can be used to more fully describe motionStill interested in displacement, velocity, and accelerationWill serve as the basis of multiple types of motion in future chapters
Position and DisplacementThe position of an object is described by its position vector, rThe displacement of the object is defined as the change in its positionr = rf - ri
General Motion IdeasIn two- or three-dimensional kinematics, everything is the same as in one-dimensional motion except that we must now use full vector notationPositive and negative signs are no longer sufficient to determine the direction
Position vector: extends from the origin of a coordinate system to the particle.
DefinitionsDisplacement vector: represents a particles position change during a certain time interval.
Average VelocityThe average velocity is the ratio of the displacement to the time interval for the displacement
The direction of the average velocity is the direction of the displacement vector, r
Average VelocityThe average velocity between points is independent of the path takenThis is because it is dependent on the displacement, also independent of the path
Instantaneous VelocityThe instantaneous velocity is the limit of the average velocity as t approaches zeroThe direction of the instantaneous velocity is along a line that is tangent to the path of the particles direction of motion
Instantaneous velocity:The direction of the instantaneous velocity of a particle is always tangent to the particles path at the particles position
Average AccelerationThe average acceleration of a particle as it moves is defined as the change in the instantaneous velocity vector divided by the time interval during which that change occurs.
Average AccelerationAs a particle moves, v can be found in different waysThe average acceleration is a vector quantity directed along v
Instantaneous AccelerationThe instantaneous acceleration is the limit of the average acceleration as t approaches zero
Producing An AccelerationVarious changes in a particles motion may produce an accelerationThe magnitude of the velocity vector may changeThe direction of the velocity vector may changeEven if the magnitude remains constantBoth may change simultaneously
Kinematic Equations for Two-Dimensional MotionWhen the two-dimensional motion has a constant acceleration, a series of equations can be developed that describe the motionThese equations will be similar to those of one-dimensional kinematics
Kinematic EquationsPosition vector
Since acceleration is constant, we can also find an expression for the velocity as a function of time: vf = vi + at
Kinematic EquationsThe velocity vector can be represented by its componentsvf is generally not along the direction of either vi or at
Kinematic EquationsThe position vector can also be expressed as a function of time:rf = ri + vit + at2This indicates that the position vector is the sum of three other vectors:The initial position vectorThe displacement resulting from vi tThe displacement resulting from at2
Kinematic EquationsThe vector representation of the position vectorrf is generally not in the same direction as vi or as airf and vf are generally not in the same direction
Kinematic Equations, ComponentsThe equations for final velocity and final position are vector equations, therefore they may also be written in component formThis shows that two-dimensional motion at constant acceleration is equivalent to two independent motionsOne motion in the x-direction and the other in the y-direction
Kinematic Equations, Component Equationsvf = vi + at becomesvxf = vxi + axt vyf = vyi + ayt rf = ri + vi t + at2 becomesxf = xi + vxi t + axt2 yf = yi + vyi t + ayt2
At t = 0, a particle moving in the xy plane with constant acceleration has a velocity of and is at the origin. At t = 3.00 s, the particle's velocity is . Find (a) the acceleration of the particle and (b) its coordinates at any time t.
Projectile MotionAn object may move in both the x and y directions simultaneouslyThe form of two-dimensional motion we will deal with is called projectile motion
Assumptions of Projectile MotionThe free-fall acceleration g is constant over the range of motionAnd is directed downwardThe effect of air friction is negligibleWith these assumptions, an object in projectile motion will follow a parabolic pathThis path is called the trajectory
Verifying the Parabolic TrajectoryReference frame choseny is vertical with upward positiveAcceleration componentsay = -g and ax = 0Initial velocity componentsvxi = vi cos q and vyi = vi sin q
II. Projectile motionHorizontal motion: ax=0 vx=v0xThe horizontal and vertical motions are independent from each other.
II. Projectile motionRange (R): horizontal distance traveled by a projectile before returning to launch height.
II. Projectile motion Vertical motion: ay= -g
Trajectory: projectiles path.We can find y as a function of x by eliminating time
Horizontal range:R = x-x0; Vertical displacement:y-y0=0. (Maximum for a launch angle of 45 )
Projectile Motion Problem Solving HintsSelect a coordinate systemResolve the initial velocity into x and y componentsAnalyze the horizontal motion using constant velocity techniquesAnalyze the vertical motion using constant acceleration techniquesRemember that both directions share the same time
A rock is thrown upward from the level ground in such a way that the maximum height of its flight is equal to its horizontal range R. (a) At what angle is the rock thrown? (b) Would your answer to part (a) be different on a different planet? (c) What is the range Rmax the rock can attain if it is launched at the same speed but at the optimal angle for maximum range?
A third baseman wishes to throw to first base, 127 feet distant. His best throwing speed is 85 mi/h. (a) If he throws the ball horizontally 3 ft above the ground, how far from first base will it hit the ground? (b) From the same initial height, at what upward angle must the third baseman throw the ball if the first baseman is to catch it 3 ft above the ground? (c) What will be the time of flight in that case?xyv0h=3ftB3B1xmax0xB1=38.7m
N7: In Galileos Two New Sciences, the author states that for elevations (angles of projection) which exceed or fall short of 45 by equal amounts, the ranges are equal Prove this statement.xv0x=R=R?y=45
A ball is tossed from an upper-story window of a building. The ball is given an initial velocity of 8.00 m/s at an angle of 20.0 below the horizontal. It strikes the ground 3.00 s later. (a) How far horizontally from the base of the building does the ball strike the ground? (b) Find the height from which the ball was thrown. (c) How long does it take the ball to reach a point 10.0 m below the level of launching?
A ball is tossed from an upper-story window of a building. The ball is given an initial velocity of 8.00 m/s at an angle of 20.0 below the horizontal. It strikes the ground 3.00 s later. (a) How far horizontally from the base of the building does the ball strike the ground? (b) Find the height from which the ball was thrown. (c) How long does it take the ball to reach a point 10.0 m below the level of launching? )
Uniform Circular MotionUniform circular motion occurs when an object moves in a circular path with a constant speed
Uniform Circular MotionUniform circular motion occurs when an object moves in a circular path with a constant speedAn acceleration exists since the direction of the motion is changing This change in velocity is related to an accelerationThe velocity vector is always tangent to the path of the object
Changing Velocity in Uniform Circular MotionThe change in the velocity vector is due to the change in direction
The vector diagram shows Dv = vf - vi
Changing Velocity in Uniform Circular MotionTwo triangles are similar, so we can write:
Dividing both parts by t and using the definitions of acceleration and velocity:
Centripetal AccelerationThe acceleration is always perpendicular to the path of the motionThe acceleration always points toward the center of the circle of motionThis acceleration is called the centripetal acceleration
Centripetal AccelerationThe magnitude of the centripetal acceleration vector is given by
The direction of the centripetal acceleration vector is always changing, to stay directed toward the center of the circle of motion
PeriodThe period, T, is the time required for one complete revolutionThe speed of the particle would be the circumference of the circle of motion divided by the periodTherefore, the period is
Tangential AccelerationThe magnitude of the velocity could also be changingIn this case, there would be a tangential acceleration
Total AccelerationThe tangential acceleration causes the change in the speed of the particleThe radial acceleration comes from a change in the direction of the velocity vector