 # Do now A B + = ? The wrong diagrams Draw the right diagram for A + B.

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do now A B + = ? The wrong diagrams Draw the right diagram for A + B

3.3 projectile motion Objectives 1. Recognize examples of projectile motion. 2. Describe the path of a projectile as a parabola. 3. Resolve vectors into their components and apply the kinematic equations to solve problems involving projectile motion.

question The long jumper builds up speed in the x-direction and jumps, so there is also a component of speed in the y direction. Does the angle of take-off matter to the jumper?

A parabola is the set of all points in the plane equidistant from a given line (the conic section directrix) and a given point not on the line (the focus). y ~ x2

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

Effect of air resistance

demo The vertical motion of a projectile is same as free falling object, with constant acceleration: a = m/s/s

Two types of projectiles
Projectile launched horizontally Projectile launched at an angle

Projectile Launched horizontally

Projectile Launched at an angle

Horizontal Projectiles
Over the Edge Horizontal Projectiles

Projectiles Launched Horizontally
Vertical motion: Free fall free rest: vy = 0 ay = m/s2 Horizontal motion: vx = vix = constant ax = 0 Thus, the projectile travels with a constant horizontal velocity and a downward vertical acceleration.

Velocity of Horizontal Projectiles
Horizontal motion is constant: velocity is constant. Vertical: same as drop the ball from rest: velocity is increasing by 9.81 m/s every second

Projectile launched horizontally
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, the projectile has no acceleration, its velocity is constant: vx = vi In vertical direction, the projectile has acceleration: a = m/s/s. Its initial vertical velocity is zero and it changes by m/s each second. Same as an object falling from rest. The horizontal motion of a projectile is independent of its vertical motion

Example An object was projected horizontally from a tall cliff. The diagram represents the path of the object, neglecting friction. Comparing the following at point A & B: Acceleration Horizontal velocity Vertical velocity

As the red ball rolls off the edge, a green
ball is dropped from rest from the same height at the same time Which one will hit the ground first? They will hit at the SAME TIME!!!

The same time?!? How?!? The green ball falls from rest
and has no initial velocity IN EITHER DIRECTION! viy and vix = 0 The red ball has an initial HORIZONTAL velocity (vix) But does not have any initial VERTICAL velocity (viy = 0) vix

We can find an object’s displacement in EITHER DIMENSION using TIME
One Dimension at a Time Both balls begin with no VERTICAL VELOCITY Both fall the same DISTANCE VERTICALLY Find time of flight by solving in the appropriate dimension We can find an object’s displacement in EITHER DIMENSION using TIME

Example #1 A bullet is fired horizontally from a gun that is 1.7 meters above the ground with a velocity of 55 meters per second. At the same time that the bullet is fired, the shooter drops an identical bullet from the same height. Which bullet hits the ground first? Both hit the ground at the same time

Equation for horizontal projectile
θ = 0 Vertical ay = m/s2 Horizontal ax = 0 d = ½ (vi + vf)t vf = vi + at d = vit + ½at2 vf2 = vi2 + 2ad viy = visinθ = 0 y = ½ (viy + vfy)t vfy = viy + ayt y = viyt + ½ayt2 vfy2 = viy2 + 2ayy vix = vicosθ = vi x = vix∙t x and y has the same t

Need time from vertical
Example #2 An airplane making a supply drop to troops behind enemy lines is flying with a speed of 300 meters per second at an altitude of 300 meters. How far from the drop zone should the aircraft drop the supplies? Need time from vertical dy = viyt + ½ ayt2 300 m = 0 + ½ (-9.81 m/s2)t2 t = 7.82 s Use time in horizontal dx = vixt + ½ axt2 dx = (300 m/s)(7.82 s) + 0 dx = 2346 m

Need time from vertical
Example #3 A stuntman jumps off the edge of a 45 meter tall building to an air mattress that has been placed on the street below at 15 meters from the edge of the building. What minimum initial velocity does he need in order to make it onto the air mattress? Need time from vertical dy = viyt + ½ ayt2 45 m = 0 + ½ (-9.81 m/s2)t2 t = 3.03 s Use time to find v v = d / t v = 15 m / 3.03 s v = 4.95 m /s

Need time from vertical
Example #4 Example #4 A CSI detective investigating an accident scene finds a car that has flown off the edge of a cliff. The car is 79 meters from the edge of the 25 meter high cliff. What was the car’s initial horizontal velocity as it went off the edge? Need time from vertical dy = viyt + ½ ayt2 25 m = 0 + ½ (-9.81 m/s2)t2 t = 2.26 s Use time in horizontal dx = vixt + ½ axt2 79 m = vix (2.26 s) + 0 vix = m/s

Example #5 The path of a stunt car driven horizontally off a cliff is represented in the diagram below. After leaving the cliff, the car falls freely to point A in 0.50 second and to point B in 1.00 second. Determine the magnitude of the horizontal component of the velocity of the car at point B. [Neglect friction.] Determine the magnitude of the vertical velocity of the car at point A. Calculate the magnitude of the vertical displacement, dy, of the car from point A to point B. [Neglect friction.]

Class work Page 101 - Sample problem 3D Page 102 – practice 3D
Answers: 0.66 m/s 4.9 m/s 7.6 m/s 5.6 m

Do now

Projectiles Launched at an Angle
Fire Away!!! Projectiles Launched at an Angle

Projectile Launched at an angle

Projectile Vector Diagram
vfy = 0 (at top) ttot

Initial velocity 12 m/s 30° Pythagorean Theorem vi2 = vix2 + viy2
What is the vertical part of the soccer ball’s initial velocity? What is the horizontal part of the soccer ball’s initial velocity? viy = vi sin θ vix = vi cos θ 12 m/s 6 m/s 30° 10.4 m/s

What do we know or assume about the vertical part of a projectile problem?
Initial vertical velocity = vi sin θ Acceleration = m/s2 Vertical speed will be 0 at the maximum height Time to top = HALF total time in the air Find time to top using final velocity equation Vertical Distance – Max height Use time to top and solve vertical distance equation

What do we know or assume about the horizontal part of a projectile problem?
Initial vertical velocity = vi cos θ Acceleration = 0 (if we assume no air resistance) Horizontal Distance – Range Use total time and solve horizontal distance equation

The symmetrical nature of a ground launched projectile
What is the acceleration at the top of the path? What is the vertical velocity at the top of the path?

Projectile launched at an angle
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, the projectile has no acceleration, its velocity is constant: vx = vicosθ In vertical direction, the projectile has acceleration: a = m/s/s. Its velocity of a projectile changes by m/s each second. Same as a free falling object. The horizontal motion of a projectile is independent of its vertical motion

A projectile is fired with initial horizontal velocity at 10
A projectile is fired with initial horizontal velocity at m/s, and vertical velocity at +20. m/s. Determine the horizontal and vertical velocity at 1 – 5 seconds after the projectile is fired. Use g = 10 m/s/s.

Time Horizontal Velocity Vertical Accelartion 0 s 1 s 2 s 3 s 4 s 5 s

Equation for projectile motion
1D equation: d = ½ (vi + vf)t vf = vi + at d = vit + ½at2 vf2 = vi2 + 2ad Vertical ay = m/s2 viy = visinθ vfy = 0 (at top) vfy = - viy (at same height) y = ½ (viy + vfy)t vfy = viy + ayt y = viyt + ½ayt2 vfy2 = viy2 + 2ayy Horizontal ax = 0 vix = vicosθ x = vix∙t

Initial Velocity Components
Since velocity is a vector quantity, vector resolution is used to determine the components of velocity. vi2 = vix2 + viy2 SOH CAH TOA sinθ = viy / vi viy = visinθ cosθ = vix / vi vix = vicosθ vi viy θ vix Special case: horizontally launched projectile: θ = 0o: viy = visinθ = 0; vix = vicosθ = vi

Examples Determine the horizontal and vertical components
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 parallel to the horizontal. A springboard diver jumps with a velocity of 10 m/s at an angle of 80 degrees to the horizontal.

example A machine fired several projectiles at the same angle, θ, above the horizontal. Each projectile was fired with a different initial velocity, vi. The graph below represents the relationship between the magnitude of the initial vertical velocity, viy, and the magnitude of the corresponding initial velocity, vi, of these projectiles. Calculate the magnitude of the initial horizontal velocity of the projectile, vix, when the magnitude of its initial velocity, vi, was 40. meters per second.    Properties    Reason    Answer    Hint

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 the horizontal displacement, the vertical displacement, the final vertical velocity, the time to reach the peak of the trajectory, the time to fall to the ground, etc.

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. Horizontal Component Vertical Component ax = 0 m/s/s vix = 25 m/s•cos45o= 17.7 m/s x = ? ay = m/s/s viy=25 m/s•sin45o= 17.7 m/s vfy = m/s ymax = ? ttotal = ?

Solve for t – use vertical information:
vfy = viy + ayt t = 3.61 s Solve for x – use horizontal information: x = vix•t + ½ •ax•t2 x = 63.8 m Solve for peak height – use vertical information: vfy2 = viy2 + 2ayypeak At the top, vfy = 0 ypeak = 15.9 m

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. Horizontal Component Vertical Component ax = 0 m/s/s vix = vi•cosθ vix = 12 m/s•cos28o vix = 10.6 m/s ttotal = 2tup x = ? ay = -9.8 m/s/s viy = vi•sinθ viy = 12 m/s•sin28o viy = 5.6 m/s vfy = -5.6 m/s y = ? tup = ? t = 1.1 s x = 12.2 m ypeak = 1.6 m

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 7.0 seconds after it was fired, what is the horizontal component of its initial velocity? what is the vertical component of its initial velocity?

An object is thrown horizontally off a cliff with an initial velocity of 5.0 meters per second.  The object strikes the ground 3.0 seconds later.  What is the vertical speed of the object as it reaches the ground? [Neglect friction.]

In the diagram, a 10.-kilogram sphere, A, is projected horizontally with a velocity of 30. meters per second due east from a height of 20. meters above level ground. At the same instant, a 20.-kilogram sphere, B , is projected horizontally with a velocity of 10. meters per second due west from a height of 80. meters above level ground. [Neglect air friction.] Initially, the spheres are separated by a horizontal distance of 100. meters. What is the horizontal separation of the spheres at the end of 1.5 seconds?

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 Projectile has maximum range when θ = 45o Max height hmax = (visinθ)2/2g As θ increases, flight height increase Max height: θ = 90o

Class work Page 103 – sample problem 3E Page 104 – practice 3E #1-5
Answers: Yes 70.3 m 20. s; 4.8 m 6.2 m/s 17.7 m/s; 6.60 m

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