10/30 do now – on a new sheet What is the difference between g and Fg? Homework: castle learning Due Today Video Project – please email me your video or upload to google classroom Vector packet correction Due Monday Newton’s Laws packet correction

Projectile Motion objectives What is a Projectile? Characteristics of a Projectile's Trajectory Describing Projectiles with Numbers Horizontal and Vertical Components of Velocity Horizontal and Vertical Components of Displacement Initial Velocity Components Horizontally Launched Projectiles - Problem-Solving Non-Horizontally Launched Projectiles - Problem-Solving

What is a projectile? An object that is launched into the air with some INITIAL VELOCITY Can be launched at ANY ANGLE In FREEFALL after launch (no outside forces except force of gravity) The path of the projectile is a PARABOLA

Free body diagram of a projectile Fg, a Fg, a Fg, a Fg, a Fg, a Fg, a Fg, a Fg, a Fg, a

Path of a projectile The path of a projectile is parabolic, or arc. The object moves vertically as well as horizontally.

Projectile Motion and Inertia Since a projectile is in free fall after it is launched, the only force acting on it is gravity, which influence the vertical motion of the projectile, causing a vertical acceleration. The horizontal motion of the projectile is the result of inertia. There is no horizontal force. The combination of inertia and gravity causes the parabolic trajectory that is characteristic of projectiles.

Effect of air resistance Ideal projectile Projectile with air resistance

Characteristics of a Projectile's Trajectory There are the two components of the projectile's motion – horizontal and vertical motion. These two perpendicular components of motion are independent of each other, which means one component does not affect the other component.

Horizontal components

Vertical components

Projectile components There are two components in projectile motion: horizontal and vertical. These two components are independent of each other. Since the only force acting on the projectile is gravity, which is in the vertical downward direction, it cause vertical acceleration only. The projectile’s vertical motion is the same as if it is in free fall with ay = -g, the vertical velocity changes by -9.81 m/s every second. There is no force acting on the projectile in horizontal direction. The horizontal acceleration is zero. The horizontal motion is only affected by inertia, which means its horizontal velocity is constant.

(Constant or Changing?) Horizontal Motion Vertical Forces (Present? - Yes or No) (If present, what dir'n?) Acceleration Velocity (Constant or Changing?) Horizontal and vertical components are independent of each other. Change of horizontal speed does not affect vertical motion. Change of vertical speed does not affect horizontal motion.

Motion graphs of projectile d vs. t graphs Horizontal motion Vertical motion dx dy t t v vs. t graphs vx vy t t

What we know about projectile motion A projectile is any object upon which the only force is gravity, projectiles travel with a parabolic trajectory due to the influence of gravity. In horizontal direction: there are no horizontal forces acting upon projectiles and thus no horizontal acceleration. The horizontal velocity of a projectile is constant (a never changing in value). In vertical direction: there is a vertical acceleration caused by gravity; its value is 9.81 m/s/s, down. The vertical velocity of a projectile changes by -9.81 m/s each second The horizontal motion of a projectile is independent of its vertical motion. When solving projectile problems, we must separate horizontal and vertical parameters. Apply equations only in horizontal or vertical direction.

Example A student throws a 5.0-newton ball straight up. What is the net force on the ball at its maximum height?

Lab – Determine Initial Velocity of a launched ball Name, Partners Names, Date Purpose: what is the initial velocity of a launched ball? Material: launcher, meter stick, metal ball, clamp Procedure: write your own procedure Data table: make your own data table Data Analysis Conclusion

Describing Projectiles with Numbers: Horizontal and Vertical Velocity Horizontal components: vertical components: Object thrown up and down. vy decreases while up, increase while down.

Example: cannon ball is launched with vix = 73. 1 m/s and viy = 19 Example: cannon ball is launched with vix = 73.1 m/s and viy = 19.62 m/s upward at time t = 0.0 s. Fill in the blanks. Time ax ay vx vy 0 s 1 s 2 s 3 s 4 s 5 s

11/2 do now Describe a projectile’s path. (1) Describe force, acceleration and velocity in the horizontal direction of a projectile. (3) Describe force, acceleration and velocity in the vertical direction of a projectile. (3) What does it mean when we say the horizontal motion of a projectile is independent of its vertical motion? (1) Homework: castle learning Due today Newton’s Laws packet correction Due Friday Newton’s Laws test correction Newton’s Laws practice packet correction

The symmetrical nature of a projectile launched at an angle vy = 0 at top vfy = -viy tup = tdown ttotal = 2∙tup

Horizontal components: Describing Projectiles With Numbers: (Horizontal and Vertical Displacement) Horizontal components: vertical components: Object thrown up then come down

Example: cannon ball is launched with vix = 73. 1 m/s and viy = 19 Example: cannon ball is launched with vix = 73.1 m/s and viy = 19.62 m/s upward at time t = 0.0 s. Fill in the blanks. Time ax ay vx vy dx dy 0 s 1 s 2 s 3 s 4 s 5 s Time ax ay vx vy dx dy 0 s 1 s 2 s 3 s 4 s 5 s The symmetrical nature of a projectile's trajectory: the vertical displacement of a projectile t seconds before reaching the peak is the same as the vertical displacement of a projectile t seconds after reaching the peak.   The horizontal distance traveled by the projectile each second is a constant value.

Horizontal and vertical components are independent of each other The horizontal and vertical motions of a projectile are independent of each other. The horizontal velocity of a projectile does not affect how far (or how fast) a projectile falls vertically. Only vertical motion parameters (initial vertical velocity, final vertical velocity, vertical acceleration) determine the vertical displacement. Only horizontal motion parameters (initial horizontal velocity, final horizontal velocity, horizontal acceleration). Determine the horizontal displacement. One of the initial steps of a projectile motion problem is to determine the components of the initial velocity.

Initial Velocity Components Since velocity is a vector quantity, vector resolution is used to determine the components of velocity. SOH CAH TOA sinθ = viy / vi viy = vi∙sinθ cosθ = vix / vi vix = vi∙cosθ In these equations, angle θ is with x axis vi viy θ vix Special case: horizontally launched projectile: θ = 0o viy = visinθ = visin0o = 0; vix = vicosθ = vicos0o = vi

Example The vector diagram below represents the velocity of a car traveling 24 meters per second 35° east of north. What is the magnitude of the component of the car’s velocity that is directed eastward?

Check your understanding Ken Fused is resolving velocity vectors into horizontal and vertical components. For each case, evaluate whether Ken's diagrams are correct or incorrect. If incorrect, explain the problem or make the correction.

Practices – determine horizontal and vertical components of velocity A water balloon is launched with a speed of 40 m/s at an angle of 60 degrees to the horizontal. A motorcycle stunt person traveling 70 mi/hr jumps off a ramp at an angle of 35 degrees to the horizontal. A springboard diver jumps with a velocity of 10 m/s at an angle of 80 degrees to the horizontal.

Why components? The point of resolving an initial velocity vector into its two components is to use the values of these two components to analyze a projectile's motion and determine such parameters as dx, dy, vfy, ttotal, tup, etc. ALWAYS USE COMPONENTS IN YOUR EQUATIONS!!!

vy = 0 at top; vfy = -viy tup = tdown ttotal = 2∙tup Determination of the TIME OF FLIGHT for projectile launched at an angle, given initial speed and angle: For a projectile with given launching angle and initial velocity, we can determine the initial horizontal and vertical velocity: viy = visinθ; vix = vicosθ Its vertical motion is the same as free fall with initial vertical velocity: vy = 0 at top; vfy = -viy tup = tdown ttotal = 2∙tup vfy = viy + a∙t 0 = viy - g∙tup If we know the initial vertical velocity, we can determine the time to reach the highest the point and the total time of flight. Time is the same for both vertical and horizontal components

Example A projectile is launched with 15 m/s -velocity at an angle of 60.° above the horizontal. What is the total flight time of projectile?

Example A projectile is launched at an angle above the ground. The horizontal component of the projectile’s velocity, vx, is initially 40. m/s. The vertical component of the projectile’s velocity, vy, is initially 30. m/s. What is the projectile’s time of flight? [Neglect friction.]

Example A football is thrown at an angle of 30.° above the horizontal. The magnitude of the horizontal component of the ball’s initial velocity is 13.0 meters per second. The magnitude of the vertical component of the ball’s initial velocity is 7.5 meters per second. [Neglect friction.] Sketch a graph representing the relationship between the horizontal displacement of the football and the time the football is in the air. [1] The football is caught at the same height from which it is thrown. Calculate the total time the football was in the air. [Show all work, including the equation and substitution with units.]

11/3 do now The vector diagram below represents the velocity of a car traveling 24 meters per second 30° east of north. What is the magnitude of the component of the car’s velocity that is directed eastward? 30o Homework: castle learning Due Friday Newton’s Laws test correction Newton’s Laws practice packet correction

Range is the longest when θ is 45o Determination of HORIZONTAL DISPLACEMENT for projectile launched at an angle, given initial speed and angle and time of flight For the given projectile, we can determine the initial horizontal and vertical velocity: viy = visinθ; vix = vicosθ For a projectile launched at an angle, its horizontal motion is constant. To determine its horizontal displacement we can use Range is the longest when θ is 45o

Example A projectile is launched with 15 m/s -velocity at an angle of 60.° above the horizontal. What is the range of the projectile?

Example A projectile is launched at an angle above the ground. The horizontal component of the projectile’s velocity, vx, is initially 40. m/s. The vertical component of the projectile’s velocity, vy, is initially 30. m/s. What is the projectile’s range? [Neglect friction.]

Determination of the PEAK HEIGHT for projectile launched at an angle, given initial speed and angle For the given projectile, we can determine the initial horizontal and vertical velocity: viy = vi∙sinθ; vix = vi∙cosθ For a projectile launched at an angle, its vertical motion is the same as free fall with initial vertical velocity: At the very top, vy = 0 vy2 = viy2 + 2ay∙y 0 = viy2 – 2g∙ypeak (ay = -g)

Example A projectile is launched with 15 m/s -velocity at an angle of 60.° above the horizontal. What is the maximum height reached by the projectile?

Example A projectile is launched at an angle above the ground. The horizontal component of the projectile’s velocity, vx, is initially 40. m/s. The vertical component of the projectile’s velocity, vy, is initially 30. m/s. What is the maximum height reached by the projectile? [Neglect friction.]

Total flight time, range, max height Time to go up t = visinθ / g As θ increases, flight time increase. Max time: θ = 90o Range Range = vi2sin2θ /g Maximum range: θ = 45o Max height hmax = (visinθ)2/2g As θ increases, flight height increase. Max height: θ = 90o

Example A football is kicked with an initial velocity of 25 m/s at an angle of 45-degrees with the horizontal. Determine the time of flight, the horizontal displacement, and the peak height of the football.

example A long jumper leaves the ground with an initial velocity of 12 m/s at an angle of 28-degrees above the horizontal. Determine the time of flight, the horizontal distance, and the peak height of the long-jumper.

Example: Fill in the blanks

11/4 do now - Fill in the blanks Homework: castle learning Due Friday Newton’s Laws test correction Newton’s Laws practice packet correction

Example - Fill in the blanks 14.9 164 2.93 5.85 42 240 19.2 16.1 1.64 3.28 80 19.7 2 4.01 19.7 3.5 0.36 0.71

Horizontally Launched Projectile special case: vix = vi, viy = 0 A projectile is launched with an initial horizontal velocity from an elevated position and follows a parabolic path to the ground. Horizontal direction ax = 0 vix = vi dx=vi∙t vertical direction ay = -g viy = 0, vy = -g∙t dy = - ½ g∙t2 Predictable unknowns include the initial speed of the projectile, the initial height of the projectile, the time of flight, and the horizontal distance of the projectile.

Example A pool ball leaves a 0.60-meter high table with an initial horizontal velocity of 2.4 m/s. Predict the time required for the pool ball to fall to the ground and the horizontal distance between the table's edge and the ball's landing location.

Example A soccer ball is kicked horizontally off a 22.0-meter high hill and lands a distance of 35.0 meters from the edge of the hill. Determine the initial horizontal velocity of the soccer ball.

example A cannon elevated at an angle of 35° to the horizontal fires a cannonball, which travels the path shown in the diagram.  [Neglect air resistance and assume the ball lands at the same height above the ground from which it was launched.] If the ball lands 7.0 × 102 meters from the cannon 14.0 seconds after it was fired, what is the horizontal component of its initial velocity? what is the vertical component of its initial velocity?

11/5 Do now A student throws a 5.0-newton ball straight up. What is the net force on the ball at its maximum height? 0.0 N 9.8 N 5.0 N 4.9 N

objectives Projectile review Due Friday Due Monday Due Tuesday projectile motion (physics classroom) Due Monday Regents Physics – Projectile Practice Due Tuesday Worksheet 1.2.3-ground launched projectile Projectile test is on Tuesday

11/6 Do now A student throws a 15.0-newton ball straight up. What is the net force on the ball at its maximum height? What is the acceleration of the ball at its maximum height? What is the velocity of the ball at its maximum height?

objectives Projectile review Due today Due Monday Due Tuesday projectile motion (physics classroom) Newton’s Laws test correction Newton’s Laws practice packet correction Due Monday Regents Physics Projectile Practice Due Tuesday Worksheet 1.2.3-ground launched projectile Projectile test is on Tuesday

11/9 do now A cannonball is fired from a cliff that is 50 meters above the ground. The cannonball is fired horizontally with a speed of 120 meters per second. Calculate the horizontal distance that the cannonball will travel.

objectives Projectile review Due today Regents Physics Projectile Practice Due tomorrow Worksheet 1.2.3-ground launched projectile Projectile test is on Tuesday Due Thursday projectile test extra credit (kinematic graphing) Due Monday Projectile project: http://www.physicsclassroom.com/shwave/projectile

11/10 objectives Projectile test Due today Worksheet 1.2.3-ground launched projectile Due Thursday projectile test extra credit (kinematic graphing) Due Monday Projectile project: http://www.physicsclassroom.com/shwave/projectile

11/10 do now A ball rolls toward the edge of a table that is 110 centimeters high and lands 2.0 meters away from its edge. Determine the speed with which the ball rolls off the edge.

11/10 objectives Projectile test Due today Worksheet 1.2.3-ground launched projectile Due Thursday projectile test extra credit (kinematic graphing) Due Monday Projectile project: http://www.physicsclassroom.com/shwave/projectile

Lab – shoot for your grade NAMES, DATE, TITLE PURPOSE How far will a horizontally launched ball land? MATERIALS launcher, meter stick, target paper PROCEDURE Write a paragraph to describe how the lab is done so that anyone reading your procedure can duplicate this lab. Include the following with your paragraph: Draw a diagram of your experiment List quantity you need to measure and the tools you use to make the measurement. Indicate these quantities in your diagram State equations you need to use to solve for the impact point distance velocity