Chapter 5 Projectile Motion

What is Projectile Motion? Nonlinear motion Motion along a curved, parabolic path Affected by gravity only (ignore air resistance) Examples are: Thrown football A package dropped out of an airplane A pitched/batted baseball An airborne snowboarder A cannonball shot by a cannon A ball rolling off the edge of a table

5.4 Projectile Motion When dealing with projectile motion, it is important to treat the motion in the x-direction separate from the motion in the y-direction What happens in the x-direction has NO effect on what happens in the y- direction and vice versa In the x-direction, a projectile is just like the horizontal motion of a ball rolling freely along a level surface without friction Moves at constant velocity because there is no force acting in this direction

5.4 Projectile Motion In the y-direction, a projectile acts just like a freely falling object Acceleration = g = -10 m/s2 In each passing second, the speed changes by 10 m/s If the projectile is rising, the speed decreases by 10 m/s in the vertical direction If the projectile is falling, the speed increases by 10 m/s in the The distance the projectile travels in each passing second increases due to it changing speed

5.4 Projectile Motion Combine constant velocity motion in the horizontal direction with accelerated motion in the vertical direction, and you get a curved path Most important, the horizontal component of motion is completely independent of the vertical component of motion

Projectile Motion: Demonstration Free Fall vs. a Horizontally Launched Projectile Horizontal projectile motion – object is given an initial push left or right Does this hold true for a projectile fired at a high rate of speed, such as from a gun?

5.4 Projectile Motion

Projectile Motion Concept Check: Describe the component of projectile motion.

5.5 Projectiles Launched Horizontally If an object is given an initial horizontal push, left or right, it will not only travel forward but it will also fall due to gravity. Therefore, tracing out a parabolic path. Examples of horizontally launched projectiles: A bullet fired from a horizontally held firearm A ball rolling off a tabletop A car driving off a cliff Something dropped out of a car window A package dropped from an airplane

5.5 Projectiles Launched Horizontally Since the initial push that launches the object is strictly left or right, it gives the object NO initial vertical component of velocity. In other words, the ball will not rise viy = 0 m/s always for a horizontally launched projectile Vertically, a horizontally launched projectile (HP) will act like an object dropped from rest

5.5 Projectiles Launched Horizontally A strobe-light photo of two balls released simultaneously – one ball drops freely while the other is projected horizontally

5.5 Projectiles Launched Horizontally There are two important things to notice in the photo of two balls falling simultaneously The ball’s horizontal component of motion remains constant. Gravity acts only downward, so the only acceleration of the ball is downward. The yellow ball travels equal horizontal distances in each time interval. Both balls fall the same vertical distance in the same time. This is because gravity acts on both objects in the same manner in the vertical direction. The vertical distance fallen has nothing to do with the horizontal component of motion

5.5 Projectiles Launched Horizontally Think! At the instant a horizontally pointed cannon is fired, a cannonball held at the cannon’s side is released and drops to the ground. Which cannonball strikes the ground first, the one fired from the cannon or the one dropped?

5.5 Projectiles Launched Horizontally Concept Check: Describe the downward motion of a horizontally launched projectile.

5.6 Projectiles Launched at an Angle Any object that is given an initial push in any direction from 0° - 90° The initial velocity therefore has a horizontal, x, component as well as a vertical, y, component. The horizontal component of motion causes the object to project forward The vertical component of motion causes the object to rise and fall Use vector resolution to determine the x and y components of the initial velocity, vx and viy

5.6 Projectiles Launched at an Angle

5.6 Projectiles Launched at an Angle Just as a one-dimensional rising and falling object reaches a speed of 0 m/s at its maximum height, so does a projectile launched at an angle Specifically, this the vertical component of the velocity The horizontal component of velocity remains constant everywhere

5.6 Projectiles Launched at an Angle Just as a one-dimensional rising an falling object achieves equal speeds at equal altitudes so does a projectile At equal altitudes, velocities have the same magnitude but opposite direction The time to go up equals the time to come down

5.6 Projectiles Launched at an Angle Think! True or False…The velocity of a projectile at its highest point is zero. Think! At what location along a projectile’s path is its velocity at a minimum?

5.6 Projectiles Launched at an Angle Understanding the Variables: Vi is the overall initial launch velocity (m/s) it acts at an angle it is NEVER zero Viy is the y-component of the initial velocity that acts strictly up or down (m/s) use vector resolution to obtain this variable it is only zero if you are calling your maximum height your initial location Vx is the x-component of the initial velocity that acts strictly left or right (m/s) this variable remains constant throughout a problem

5.6 Projectiles Launched at an Angle Understanding the Variables Vf is the overall final velocity of the object (m/s) It acts at an angle It is NEVER zero Vfy is the y-component of the final velocity that acts strictly up or down (m/s) use vector resolution to obtain this variable it is only zero if you are calling your maximum height your final location

5.6 Projectiles Launched at an Angle Understanding the Variables ∆x The horizontal distance traveled between your initial and final location (m) Also referred to as “range” of a projectile ∆y The vertical distance traveled between your initial and final location (m) Sometimes may be referred to as “altitude”

Initial Locationcation Final Location vi vx viy vfy vf ∆y ∆x

5.6 Projectiles Launched at an Angle Projectiles that are launched at the same speed but at different angles reach different heights above the ground and also travel different horizontal distances.

5.6 Projectiles Launched at an Angle The same range is obtained for two different projection angles – angles that add up to 90° An object launched at an angle of 30° with travel the same horizontal distance as one launched at 60° Maximum range is usually attained at angle of 45°

5.6 Projectiles Launched at an Angle In the presence of air resistance, the path of a high-speed projectile falls below the idealized parabola and follows the solid curve

5.6 Projectiles Launched at an Angle Think! A projectile is launched at an angle into the air. Neglecting air resistance, what is its vertical acceleration? Its horizontal acceleration?

Projectile Motion Key Ideas Projectiles ALWAYS maintain a constant horizontal velocity Projectiles ALWAYS experience a constant vertical acceleration of g = -10 m/s2 Horizontal and vertical motion are completely independent of each other. For a projectile beginning and ending at the same height, the time it takes to rise to its highest point equals the time it takes to fall from the highest point back to the original position

Projectile Motion Key Ideas You must consider horizontal and vertical motion separate when working with equations and problem solving. All equations for linear motion (chapter 4) can be used for projectile motion X direction  𝑣 𝑥 = ∆𝑥 ∆𝑡 Y direction  ∆𝑦= 𝑣 𝑖𝑦 ∆𝑡 + 1 2 𝑔∆ 𝑡 2 𝑣 𝑓𝑦 2 = 𝑣 𝑖𝑦 2 +2𝑔∆𝑦 𝑣 𝑓𝑦 = 𝑣 𝑖𝑦 +𝑔∆𝑡

Common Motion Misconceptions An object dropped from a moving carrier continues to mover forward as it falls. It does NOT fall straight down or backward A projectile launched horizontally at a high rate of speed will be in the air just as long as one launched at a lower rate of speed When released from the same height, a dropped object and a horizontally launched object will strike the ground at the exact same time