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**Introduction to Projectile Motion**

Mr. Chin-Sung Lin

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**Introduction to Projectile Motion**

What is Projectile Motion? Trajectory of a Projectile Calculation of Projectile Motion

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**Introduction to Projectile Motion**

What is Projectile Motion? Trajectory of a Projectile Calculation of Projectile Motion

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**What is Projectile Motion?**

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**Features of Projectile Motion?**

Thrown into the Air 2-D Motion Parabolic Path Affected by Gravity Determined by Initial Velocity

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**Definition: Projectile Motion**

Projectile motion refers to the 2-D motion of an object that is given an initial velocity and projected into the air at an angle. The only force acting upon the object is gravity. It follows a parabolic path determined by the effect of the initial velocity and gravitational acceleration.

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**Definition: Projectile Motion**

Projectile motion refers to the 2-D motion of an object that is given an initial velocity and projected into the air at an angle. The only force acting upon the object is gravity. It follows a parabolic path determined by the effect of the initial velocity and gravitational acceleration. 7

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**Introduction to Projectile Motion**

What is Projectile Motion? Trajectory of a Projectile Calculation of Projectile Motion

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**Trajectory (Path) of a Projectile**

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**Trajectory (Path) of a Projectile**

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y v0 x

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y x

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y x

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y x

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**g = 9.81m/s2 y x Velocity is changing and the motion is accelerated**

The horizontal component of velocity (vx) is constant Acceleration from the vertical component of velocity (vy) Acceleration due to gravity is constant, and downward a = - g = m/s2 g = 9.81m/s2 x

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y The horizontal and vertical motions are independent of each other Both motions share the same time (t) The horizontal velocity vx = v0 The horizontal distance .... dx = vx t The vertical velocity vy = - g t The vertical distance dy = 1/2 g t2 g = 9.81m/s2 x

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**Trajectory (Path) of a Projectile**

The path of a projectile is the result of the simultaneous effect of the H & V components of its motion H component constant velocity motion V component accelerated downward motion H & V motions are independent H & V motions share the same time t The projectile flight time t is determined by the V component of its motion

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**Trajectory (Path) of a Projectile**

H velocity is constant vx = v0 V velocity is changing vy = - g t H range: dx = v0 t V distance: dy = 1/2 g t2

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**Introduction to Projectile Motion**

What is Projectile Motion? Trajectory of a Projectile Calculation of Projectile Motion

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**Calculation of Projectile Motion**

Example: A projectile was fired with initial velocity v0 horizontally from a cliff d meters above the ground. Calculate the horizontal range R of the projectile. g R d v0 t

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**Strategies of Solving Projectile Problems**

H & V motions can be calculated independently H & V kinematics equations share the same variable t g R d v0 t

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**Strategies of Solving Projectile Problems**

H motion: dx = vx t R = v0 t V motion: dy = d = 1/2 g t t = sqrt(2d/g) So, R = v0 t = v0 * sqrt(2d/g) g R d v0 t

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**Numerical Example of Projectile Motion**

H motion: dx = vx t R = v0 t = 10 t V motion: dy = d = 1/2 g t t = sqrt(2 *19.62/9.81) = 2 s So, R = v0 t = v0 * sqrt(2d/g) = 10 * 2 = 20 m V0 = 10 m/s g = 9.81 m/s2 19.62 m t R

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**Exercise 1: Projectile Problem**

A projectile was fired with initial velocity 10 m/s horizontally from a cliff. If the horizontal range of the projectile is 20 m, calculate the height d of the cliff. g = 9.81 m/s2 20 m d V0 = 10 m/s t

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**Exercise 1: Projectile Problem**

H motion: dx = vx t = v0 t = 10 t t = 2 s V motion: dy = d = 1/2 g t2 = 1/2 (9.81) 22 = m So, d = m g = 9.81 m/s2 20 m d V0 = 10 m/s t

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**Exercise 2: Projectile Problem**

A projectile was fired horizontally from a cliff m above the ground. If the horizontal range of the projectile is 20 m, calculate the initial velocity v0 of the projectile. g = 9.81 m/s2 20 m 19.62 m V0 t

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**Exercise 2: Projectile Problem**

H motion: dx = vx t = v0 t V motion: dy = d = 1/2 g t t = sqrt(2 *19.62/9.81) = 2 s So, = v0 t = 2 v v0 = 20/2 = 10 m/s g = 9.81 m/s2 20 m 19.62 m V0 t

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**Summary of Projectile Motion**

What is Projectile Motion? Trajectory of a Projectile Calculation of Projectile Motion

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**Projectile Motion with Angles**

Mr. Chin-Sung Lin

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**Example: Projectile Problem – H & V**

A projectile was fired from ground with 20 m/s initial velocity at 60-degree angle. What’s the horizontal and vertical components of the initial velocity? g = 9.81 m/s2 20 m/s vy 60o vx

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**Example: Projectile Problem – At the Top**

A projectile was fired from ground with 20 m/s initial velocity at 60-degree angle. What’s the velocity of the projectile at the top of its trajectory? v g = 9.81 m/s2 20 m/s vy t 60o vx R

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**Example: Projectile Problem – Height**

A projectile was fired from ground with 20 m/s initial velocity at 60-degree angle. What’s the maximum height that the ball can reach? g = 9.81 m/s2 20 m/s vy h 60o vx

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**Example: Projectile Problem - Time**

A projectile was fired from ground with 20 m/s initial velocity at 60-degree angle. How long will the ball travel before hitting the ground? g = 9.81 m/s2 20 m/s vy t 60o vx

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**Example: Projectile Problem – H Range**

A projectile was fired from ground with 20 m/s initial velocity at 60-degree angle. How far will the ball reach horizontally? g = 9.81 m/s2 20 m/s vy 60o vx R

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**Example: Projectile Problem – Final V**

A projectile was fired from ground with 20 m/s initial velocity at 60-degree angle. What’s the final velocity of the projectile right before hitting the ground? g = 9.81 m/s2 20 m/s vy 60o vfx vx vfy vf

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**Example: Projectile Problem – Max R**

A projectile was fired from ground with 20 m/s initial velocity. How can the projectile reach the maximum horizontal range? What’s the maximum horizontal range it can reach? g = 9.81 m/s2 20 m/s q R

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