Four Major Types of Two Dimensional Motion 1. Projectile Motion 2. Circular Motion 3. Rotational Motion 4. Periodic Motion

Projectile motion problems are best solved by treating horizontal and vertical motion separately. IMPORTANT Gravity only affects vertical motion. There are two general types of projectile motion situations. 1. o oo object launched horizontally 2. o oo object launched at an angle

Object Launched Horizontally v x = initial horizontal velocity R x = horizontal range h = initial height above ground t = total time in the air IMPORTANT FACTS There is no horizontal acceleration. There is no initial vertical velocity. The horizontal velocity is constant. Time is the same for both vertical and horizontal. horizontal Rx = vxt vertical h = 0.5gt2

Object Launched at an Angle v = initial velocity  = launch angle h = maximum height t = total time in air R x = horizontal range IMPORTANT FACTS The horizontal velocity is constant. It rises and falls in equal time intervals. It reaches maximum height in half the total time. Gravity only effects the vertical motion. horizontal vx = v cos Rx = vxt vertical vy = v sin h = vyt/4 t = 2vy/g

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constant initial velocity versus how the horizontal range changes with angle; plot “range vs angle”constant initial velocity versus how the horizontal range changes with angle; plot “range vs angle” constant initial velocity versus how total time in air changes with angle; plot “total time vs angle”constant initial velocity versus how total time in air changes with angle; plot “total time vs angle” constant initial velocity versus how maximum height changes with angle; plot “height vs angle”constant initial velocity versus how maximum height changes with angle; plot “height vs angle” constant angle versus how the horizontal range changes with initial velocity; plot “range vs velocity”constant angle versus how the horizontal range changes with initial velocity; plot “range vs velocity” constant angle versus how the total time in the air changes with initial velocity; plot “time vs velocity”constant angle versus how the total time in the air changes with initial velocity; plot “time vs velocity” constant angle versus how the maximum height changes with initial velocity; plot “height vs velocity”constant angle versus how the maximum height changes with initial velocity; plot “height vs velocity” Suggested Constructivist Activities Students use simulations to complete data tables and make graphs of the following situations:

object moves in circular path about an  point (“revolves”)

According to Newton’s First Law of Motion, objects move in a straight line unless a force makes them turn. An external force is necessary to make an object follow a circular path. This force is called a CENTRIPETAL CENTRIPETAL (“center (“center seeking”) seeking”) FORCE. Since every every unbalanced force causes an object to accelerate in the direction of that force (Newton’s (Newton’s Second Law), Law), a centripetal force causes a CENTRIPETAL ACCELERATION. ACCELERATION. This acceleration results from a change in direction, and does not imply a change in speed, although speed may also change.

Centripetal force and acceleration may be caused by: gravity - planets orbiting the sungravity - planets orbiting the sun friction - car rounding a curvefriction - car rounding a curve a rope or cord - swinging a mass on a stringa rope or cord - swinging a mass on a string r m In all cases, a mass m moves in a circular path of radius r with a linear speed v. The time to make one complete revolution is known as the period, T. v The speed v is the circumference divided by the period. v = 2r/T

The formula for centripetal acceleration is: ac = v2/r and centripetal force is: Fc = mac = mv2/r m = mass in kg v = linear velocity in m/s F c = centripetal force in N r = radius of curvature in m a c = centripetal acceleration in m/s 2

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object moves in circular path about an  point or axis (“rotates” or “spins”)

The amount that an object rotates is its angular displacement. angular displacement, , is given in degrees, radians, or rotations. 1 rotation = 360 deg = 2 radians The time rate change of an object’s angular displacement i ii is its angular velocity. angular velocity, , is given in deg/s, rad/s, rpm, etc...

The time rate change of an object’s angular velocity is its angular acceleration. Angular acceleration, , is given in deg/s2, rad/s2, rpm/s, etc... Formulas for rotational motion follow an exact parallel with linear motion formulas. The only difference is a change in variables and a slight change in their meanings.

Constant LINEAR v f = v i + at d = v av t v av = (v f + v i )/2 d = v i t + 0.5at 2 v f 2 = v i 2 + 2ad ROTATIONAL  f  =  i +  t  =  av t v av =  (  f  +  i )/2  =  i t  +  0.5  t 2  f 2  =  i 2  +  2 

PERODIC MOTION any motion in which the path of the object repeats itself in equal time intervals The simple pendulum is a great example of this type of motion.

The period, T, of a simple pendulum (time needed for one complete cycle) is approximated by the equation: where l is the length of the pendulum and g is the acceleration of gravity.

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