2Projectile MotionA projectile is any object which once a force is used to throw it, hit it , propel it in some fashion, no other force acts on the object except for gravity.Projection can behorizontal, with no initial vertical velocityvertical, with some initial vertical velocity
3Projectile Motion: A special case of uniformly accelerated motion This path is a parabola.Projectile Motion: A special case of uniformly accelerated motionIf air resistance is negligible then only gravityaffects the path (trajectory) of a projectile.
4Characteristics of Projectile Motion The motion’s dimensions are vectors.The projectile will move along it’s trajectory (path) in both the horizontal (x) direction and the vertical (y) direction at the same time.The two directions are independent of each other.Only time is the same between the horizontal and vertical motion. It is, after all, only one object.
6Real Motion is the Combination of the Horizontal and Vertical Motions The blue dot show the real motionThe path taken is the trajectoryThis is horizontal projection
7Initial Velocity – V Along the Trajectory Angular projection - The initial velocity is the resultant of adding the two vector quantities together.The projection includes an angle of projection
8Initial Velocity has an X and Y Component The vertical component and the horizontal component are independent of each other.
9Horizontal and Vertical Components of Velocity Vertical velocity decreases at a constant ratedue to the influence of gravity.It becomes zero.Then increases in the negative directionNegative velocitygets largerVertical velocity = 0Positive velocitygets smaller
10Calculating Components You have learned to calculate components of a vector when we looked at inclined planes.The components are calculated by using the trig functions.The initial velocity acts as the hypotenuse of the right triangle.Vi
11Continued Calculation The vertical velocity and the horizontal velocity are the legs of the triangle.VyVxViCalculate the Vy using sin = o/hCalculate Vx using cos = a/h
12Horizontal Displacement of Projectiles Horizontal projection or projection at an anglegraph of horizontal displacement v timeAs time progresses, the object gets further and further from its starting point.TimeDisplacement
13Vertical Displacement of Projectiles Vertical Projection and Projection at an Anglegraph of vertical displacement (height) v timeThe object rises into the air and then return to earth.TimeHeight
14Vertical Displacement of Projectiles Horizontal projectionGraph of vertical displacement (height) vs timeThe object leaves a horizontal surface and fall to the ground.TimeHeight
15Horizontal Velocity of Projectiles Horizontal Projection and Projection at an AngleTimeVelocityThe horizontal velocity of a projectile remains constant from the time it is projected until gravity brings it to the ground.Remember: We are using an ideal situation where there is no air resistance
16Vertical Velocity of Projectiles Vertical Projection and Projection at an AngleTimeVelocityFor objects projected directly upward or projected at some angle above the ground, the vertical velocity must begin positive, decrease to zero, and then increase in the negative direction. (Remember, gravity is negative)
17Vertical Velocity of Projectiles Horizontal projectionTimeVelocityHorizontal projection begins with an initial vertical velocity of zero.The vertical velocity then increases in a negative direction.
18Horizontal Acceleration of Projectiles Since we are idealizing the projection, we do not take into account any air resistance.We can, therefore, say there is no horizontal acceleration.TimeAcceleration
19Vertical Acceleration of Projectiles Vertical acceleration is the result of the pull of gravity, (-9.8 m/s2)This is the same on the way up and on the way down.Acceleration-9.8Time
20Important Notes on Vertical and Angular Projection The range (x) is the farthest the object will travel horizontally.The maximum height (ymax ) is the farthest the object will travel vertically.Y equals zero when it is at its lowest point.
21Determining the RangeYou can determine the displacement (range) of a projectile any any point along the trajectory.X = horizontal distance (range)vx = horizontal velocityt = time
22Determining the Height You can determine the height (y) at any point in the trajectory!y = vertical displacementvy = initial vertical velocityg = acceleration due to gravityt = time
23Other Important Notes on Vertical and Angular Projection At the highest point of the trajectory, it is the exact midpoint of the time.It takes the projectile half of the time to get to the top.When the projectile gets to the top, it has to stop going up and start going down, so the velocity in the y-direction at the highest point is zero for a split second.As the projectile falls, it is in free fall.
243 Primary Factors Affecting Trajectory Projection angleaka release angle or take-off angleProjection velocityaka initial or take-off velocityProjection heightaka above or below landing
25Projection AngleThe optimal angle ofprojection is dependent onthe goal of the activity.For maximum height, the optimal angle is 90o.For maximum distance, the optimal angle is 45o.
26The effect of Projection angle on the Range of a projectile 10 degrees30 degrees40 degrees45 degrees60 degrees75 degreesThe angle that maximizes Range is = 45 degrees
27The effect of Projection velocity on the Range of a projectile 10 45 degrees Range ~ 10 m20 45 degrees Range ~ 40 m30 45 degrees Range ~ 90 m100908070605040302010
28Projectile ProblemsIgnore air resistance.ay = g = m/s2Set up the two dimension separatelyOrigin x Origin yPositive x Positive yxi = constant yi =vxi = vyi =ax = ay = g
29Projectile Problems – Two Dimensional Kinematics Write general kinematic equations for each directionRewrite them for the problem at handFind the condition that couples the horizontal and vertical motions (usually time)
30Equations of Constant Acceleration This is the only equation to use for the horizontal part of the motionx = vxtd = ½ (vyf + vyi)td = vyit + ½ gt2vvf2 = vvi2 + 2gdThese 3 equations are for the vertical part of the motion
31The Monkey and the Banana A zookeeper must throw a banana to a monkey hanging from the limb of a tree. The monkey has a habit of dropping from the tree the moment that the banana is thrown. If the monkey lets go of the tree the moment that the banana is thrown,will the banana hit the monkey?
32When you take gravity into consideration you STILL aim at the monkey! Monkey’sGravity free path is “floating” at height of limbBanana’sGravity free pathFall thru same heightIt works! Since both banana and monkey experience the same acceleration each will fall equal amounts below their gravity-free path. Thus, the banana hits the monkey.