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Aim: How can we approach projectile problems? Do Now: An object is launched up vertically with an initial velocity of v oy and lands back on the ground in time t. Using the formula, d = v oy t + ½ at 2, how long will it be in the air for?

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Separate Components Velocity, acceleration, and distance have both horizontal (x) component, and vertical (y) component

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Horizontally Fired Objects Horizontal velocity is constant: There is no horizontal acceleration: Horizontal distance (also known as the range) is found using this equation: a = 0 m/s 2

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Vertical Components For the vertical motion, the object is in free fall: Vertical acceleration = gravity: Vertical distance (also known as the height) is found using the distance equation: v o = 0 m/s a = g = 9.8 m/s 2 y = v o t + ½at 2

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Projectiles Fired at an Angle Unlike horizontal projectiles, there are both v ox and v oy Initial velocity can be resolved into x and y components vivi v ox = v o cosθ v oy = v o sinθ θ

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Now what? Since you can determine your v ox and v oy, the rest we can figure out!!! Horizontal velocity is still constant Horizontal acceleration is still zero Vertical acceleration is -10 m/s 2 (just like in vertically fired objects) Vertical v f at max height is 0 m/s (just like in vertically fired objects) Vertical displacement is 0 m (the object starts on the ground and ends on the ground) Time to max height = ½ total time

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Time Vertical and Horizontal time are equal:

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1. An object is released from rest on a planet that has no atmosphere. The object falls freely for 3.0 meters in the first second. What is the magnitude of the acceleration due to gravity on the planet? (A) 1.5 m s 2 (B) 3.0 m/s 2 (C) 6.0 m/s 2 (D) 10.0 m/s 2 (E) 12.0 m/s 2 No Calculator Allowed **1 minute**

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2. An object is thrown with a horizontal velocity of 20 m/s from a cliff that is 125 m above level ground. If air resistance is negligible, the time that it takes for the object to fall to the ground from the cliff is most nearly (A)3 s (B)5 s (C)6 s (D)12 s (E)25 s No Calculator Allowed **1 minute**

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3. A 2 ‑ kilogram block rests at the edge of a platform that is 10 meters above level ground. The block is launched horizontally from the edge of the platform with an initial speed of 3 meters per second. Air resistance is negligible. The time it will take for the block to reach the ground is most nearly (A)0.3 s (B)1.0 s (C)1.4 s (D)2.0 s (E)3.0 s No Calculator Allowed **1 minute**

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4. A diver initially moving horizontally with speed v dives off the edge of a vertical cliff and lands in the water a distance d from the base of the cliff. How far from the base of the cliff would the diver have landed if the diver initially had been moving horizontally with speed 2v? (A)D (B) (C)2d (D)4d (E) can’t be determined without knowing the height of the cliff Velocity is directly proportional to distance 2v = 2d No Calculator Allowed **1 minute**

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5. A projectile is fired with initial velocity v o at an angle 0 with the horizontal and follows the trajectory shown above. Which of the following pairs of graphs best represents the vertical components of the velocity and acceleration v and a, respectively, of the projectile as functions of time t? No Calculator Allowed **1 minute** Vertical acceleration is constant (-10 m/s 2 )

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Questions 6-7 A ball is thrown and follows the parabolic path shown above. Air friction is negligible. Point Q is the highest point on the path. Points P and R are the same height above the ground. 6. How do the speeds of the ball at the three points compare (A) v p < v Q < v R (B) v R < v Q < v P (C) v Q < v R < v P (D) v Q < v P = v R (E) v P = v R < v Q No Calculator Allowed **1 minute** Points P and R have the same height, therefore the same speed (velocity is different) Point Q has less speed (x-comp is the same but there is no y-comp) Resultant is less

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7. Which of the following diagrams best shows the direction of the acceleration of the ball at point P ? (A) (B) (C) (D) (E) Acceleration is due to gravity Gravity always go down No Calculator Allowed **1 minute**

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8. How much time does it take the rock to travel from the edge of the building to the ground? (A) (B) (C) (D) (E) A rock of mass m is thrown horizontally off a building from a height h, as shown above. The speed of the rock as it leaves the thrower’s hand at the edge of the building is v 0. No Calculator Allowed **1 minute** Or try dimensional analysis

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9. A 0.50 kg cart moves on a straight horizontal track. The graph of velocity v versus time t for the cart is given below. From t = 25 s until the cart reaches the end of the track, the cart continues with constant horizontal velocity. The cart leaves the end of the track and hits the floor, which is 0.40 m below the track. Neglecting air resistance, determine each of the following: i. The time from when the cart leaves the track until it first hits the floor ii. The horizontal distance from the end of the track to the point at which the cart first hits the floor Calculator Allowed **2.5 minutes**

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