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Projectile Motion

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**Projectile An object that is moving in two directions**

The directions are horizontal and vertical Also called X and Y on a graph Has a single force that acts upon it - the force of gravity

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Motion of a projectile Forces are only required to cause an acceleration (not a motion) Inertia causes an object to continue to move in a straight line unless acted on by an outside force There is a downward force and a downward acceleration If shot upward, the object will be moving upward and slowing down

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Components of motion There are two components of motion – horizontal and vertical Only the vertical motion is affected by gravity The two components of motion are independent of each other The object will continue to move horizontally at constant velocity until it hits the ground The distance the object travels is the velocity times the time in the air Horizontal distance comes from horizontal velocity, this velocity is constant, so it uses a linear formula. There is no acceleration, so the formula is actually from velocity equals distance divided by time

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**The path is curved Gravity is an acceleration**

Acceleration increases velocity The object moves faster and faster The object moves a longer and longer distance in a given period of time The shape given by this type of movement is a parabola Projectiles travel with a parabolic trajectory due to the fact that the downward force of gravity accelerates them downward from their otherwise straight-line, gravity-free trajectory

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**Two components of motion**

The two components are independent, they have to be calculated independently The initial velocity is a vector, which has a magnitude and a direction The X and Y components of the initial velocity vector form a right triangle VX = V cos θ VY = V sin θ

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Calculations Given a velocity and an angle, solve for the distance traveled Sample problem: V = 35 m/s, θ = 40° Step 1: VX = V cos θ = 35 cos 40° = 26.8 m/s Step 2: VY = V sin θ = 35 sin 40° = 22.5 m/s Distance equals velocity times time Distance is an X value, use VX or D = VX X t You still need to calculate time VX is velocity in the X direction, VY is velocity in the Y direction

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**Calculating time Calculating time is harder, it has two parts**

The time it takes to go up The time it takes to come down Total time is the sum of both, or t = t1 + t2 Each of the two times is calculated separately

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Time One

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Time One (alternate)

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**t = t1 + t2 = 2.3 s + 2.3 s = 4.6 sec Time Two**

Time two comes from the time it takes the object to fall from its peak height If the object starts at the same height it ends up at, then the time to come down is the same as the time to go up t = t1 + t2 = 2.3 s s = 4.6 sec Now you can solve for distance traveled: D = VX X t = 26.8 m/s x 4.6 s = m

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Time Two continued

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**Calculate distance First calculate total time**

t = t1 + t2 = 2.3 s s = 5.68 s Now calculate distance traveled D = VX X t = 26.8 m/s x 5.68 s = m

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Alternate Methods There are several other ways to work projectile motion problems One of the ways is to solve for time using a quadratic equation

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