PH 201 Dr. Cecilia Vogel Lecture 4

REVIEW  Constant acceleration  x vs t, v vs t, v vs x  Vectors  notation  magnitude and direction OUTLINE  2-D motion with acceleration  Projectiles  acceleration of gravity  Circular motion  constant speed with acceleration

Special Case: Projectile Motion  Object moving with no acceleration except that of gravity.  falling object  thrown object  This is 2-D motion, so vector equations stand for  the y- motion is  a y = -g = -9.8 m/s 2  g is a positive #  the x-motion is  a x =

Projectile Motion x-component y-component

Special Case:  How long will it take a thrown object to reach its max height, h? Given v oy = vo*sin(theta) y i = 0 At max height, v y =0 find t Note – time to go up and back down is twice that

Special Case:  How high will thrown object go? v oy = vo*sin(theta) y i = 0 At max height, v y =0, y=h find h

Range  Height and time only depend on y-component of initial velocity!!!!!  Range = horizontal distance covered while going up and back down to original height  Range depends on both components  Horizontal motion is constant velocity, so  R= x = v ox t  where t=time up and back down

Special Case: Uniform Circular Motion   is the (constant) “angular velocity”  positive if CCW  So x and y change sinusoidally  Direction angle changes at a constant rate x-axis 

Period of Motion  Period, T, is the time it takes  time to go  If it goes all the way around once, the angle changes by (rad)  If it goes all the way around once, the distance traveled is

Special Case: Uniform Circular Motion  An object moves in a circle of constant radius, r, with constant speed, v.  Is the object accelerating?   Consider ways of ID’ing acceleration:  physical intuition: force needed, jerk felt  math:

Acceleration  What is the acceleration of the object at  Consider the average acceleration from just before to just after:  Generally: centripetal acceleration is

Magnitude of Acceleration  Same r and larger v yields:  larger accel  Same v and smaller r yields:  larger accel  Same T and larger r yields:  larger accel  even tho r is larger, v is, too!

Example  What is the speed and acceleration of the Earth in orbit?  r = 93,000,000 mi = 1.5X10 11 m  T = 1yr = 3.156X10 7 s = 30,000 m/s = m/s 2 huge circle –

Summary  2-D accelerated motion  Projectile motion  constant horizontal speed  gravitational acceleration is vertical  path is part of a parabola  special case of dropped object  Uniform circular motion  constant speed, changing direction  velocity tangent to circle  acceleration toward center of circle