chapter 4 motion in 2 dimensions. overview the focus of this chapter is kinematics in 2-d –...
TRANSCRIPT
Chapter 4
Motion in 2 Dimensions
Overview
• The focus of this chapter is kinematics in 2-D– Projectile Motion– Uniform Circular Motion– Tangential/Radial Acceleration– Relative Motion
4.1 Pos, Vel, Accel Vectors
• Extending what we know about 1-D (straight line) motion to 2-D (motion in xy plane)
• r - position vector (x i + y j )(Points from the origin)
• Δr = rf - ri
displacement vector
(vector subtraction = tail to tail)
4.1
• - Average Velocity Vector(points along direction of Δr, t is a scalar)
• - Instantaneous Velocity
– First Derivative of the Position Vector Function with respect to time
t
r
v
dt
d
tt
rrv
0lim
4.1
4.1
• Acceleration- rate of change of velocity
• - average acceleration
• - instantaneous acceleration
– First Derivative of the Velocity Vector Function– Second Derivative the Position Vector Function
if
if
ttt
vvv
a
dt
d
tt
vva
0lim
4.1
• Remember- acceleration = rate of change of vAccel can be cause by changes in
Magnitude (speed)Direction
• Quick Quizzes Pg 80
4.2 2-D Motion with cons. Accel
• We can study an object moving in two dimensions if its position vector as a function of time is know.
• - Position function
• - Velocity function
jir ˆ)(ˆ)( tytx
jir
v ˆˆdt
dy
dt
dx
dt
d
jiv ˆˆyx vv
4.2
• Example 4.1 Pg 82
4.3 Projectile Motion
• Projectile Motion– Easily studied with two assumptions.• Vertical Motion is equivalent to free fall (-g)• Air Resistance is Negligible
– The path of the projectile (trajectory) is parabolic in shape.
– Track the motion as two separate functions• Up and Down (free fall)• Left and Right (uniform motion)
4.3
4.3
• Quick Quizzes Pg 85• Example 4.2 Pg 85
• Vertical Height (See Board Derivation)• Horizontal Range (see Board Derivation)• Maximum Range (45o)
4.3
• Example Problems 4.4-4.7
4.4 Uniform Circular Motion
• An object following a circular path at constant speed.
• Acceleration is due to changing direcition of the tangent velocity vector.
4.4
• Centripetal Acceleration– Points to the Center of the Circular Path– Perpendicular to the tangent velocity
Quick Quizzes/Example Pg 93
2
C
va
r
T
rv
2 rfv 2
4.5 Tangential and Radial Accel
• If the speed of an object is not constant around a circular path– The portion of the acceleration due to changing
direction- radial acceleration– The portion of the acceleration due to changing
speed- tangential acceleration
4.5
4.5
• Tang. Accel
• Radial Accel
• Total Accel (Magnitude)
• Total Unit Vector Accel
t
da
dt
v
2
r C
va a
r
2 2r ta a a
2ˆ ˆt r
d v
dt r
va a a r
4.5
2ˆ ˆt r
d v
dt r
va a a r
4.5
• Remember- for uniform circular motionat = 0
Quick Quizzes/Example Pg 95
4.6 Relative Velocity and Accel
• How motion is observed from a moving frame of reference rather than fixed frame.
• Airport People Movers (moving sidewalk) Fig 4.21
• Ball and Skateboard Fig 4.22
4.6
• The observed displacements and velocities are different to the two observers
• The accelerations however remain the same assuming that the moving frame has constant speed.
4.6
Quick Quiz Pg 98Examples 4.10, 4.11 Pg 98-99