Unit 1c: Projectile motion! Why is it worse to fall from greater heights (without a parachute)? Acceleration due to gravity β’ On earth, in a vacuum, all objects fall with the same acceleration due to gravity: 9.81 m/s2 downward β’ Two objects of different weight (e.g., feather and bowling ball) dropped from the same height will land at the same time How long does it take to hit the ground? 1 2 β’ π = β ππ‘ 2 β’ A rock is dropped from a height of 100.0 m. Ignoring air resistance, how long will it take to hit the ground? Letβs test it! Trial Time 1 Time 2 Time 3 Averag e time Movement in two dimensions β’ Vectors (in two dimensions) can be broken down into vector components of each dimension. β’ Each component vector (for say, velocity) behaves and can be analyzed independently. β’ Plain English: the vertical velocity is not influenced by the horizontal velocity, and vice versa. Vy (vertical velocity) Vx (horizontal velocity) Horizontally launched projectiles β’ The horizontal velocity should not change (ignoring air resistance) β’ The vertical velocity changes due to gravitational acceleration β’ KNOW THIS: A ball rolling off of a table should hit the floor at the same time as ball that was dropped from the same height simultaneously. How to solve a horizontally launched projectile problem β’ Use the equations: 1 2 β’ ππ¦ = β ππ‘ 2 for vertical distance or time β’ ππ₯ = π£π₯ β π‘ for horizontal distance traveled, once you know t (time it took for object to fall certain vertical distance). Practice β’ A ball rolls off of a 1.00 meter-high table at a velocity of 1.5 m/s. How far away from the table will it land? β’ Step 1: solve for time to hit ground using first equation β’ Step 2: solve for horizontal distance traveled using second equation What about projectiles launched at an angle??? β’ The initial vertical velocity is not zero. Projectile basics: β’ An objected projected upward at an angle will (neglecting air resistance)... β’ have a vertical velocity of zero at its maximum height β’ Have a constant horizontal velocity β’ always be accelerating downward (-9.81 m/s2) β’ take the same amount of time to reach maximum height as it will to fall down from that maximum height β’ Will have the opposite vertical velocity (same speed, opposite direction) when it falls to its original height Practice concepts β’ At ground level a cannon ball is shot upwards at an angle of 45 degrees. It takes 10 seconds for it to reach maximum height. How long will the cannon ball be in the air, from the beginning? β’ A ball is tossed up at an angle with a vertical velocity of 3 m/s upward. What will its velocity be when it hits the ground? What about calculating distance and time based off of intial velocity? β’ First, We need to break down a vector into its horizontal and vertical componentsβ¦ To break a vector down into its dimensional componentsβ¦ β’ Draw a horizontal vector pointing in the same direction of the original vector β’ Draw a vertical vector pointing in the same direction β’ Make a right triangle. The original vector is the hypotenuse How to calculate the magnitude of the component vectors β’ To find vx (horizontal vector), multiply the magnitude of the original vector by the cosine of the angle of the vector. β’ π£π₯ = π£ β πππ π β’ To find vy (vertical component), multiply the magnitude of the original vector by the sine of the angle of the vector β’ π£π¦ = π£ β π πππ Vy = 10 m/s * sin (30°) Vx = 10 m/s * cos (30°) β’ Once you know the vertical velocity (π£π¦ ), you may use the rearranged acceleration equation to find how long it takes for object to reach peak height, when vertical velocity is zero: β’ vf = 0 = vi + at β’ You can find total time by multiplying time to peak height by 2 β’ Alternatively, you can find the vertical displacement (how high), or time (how long to get there), or initial velocity using this equation: 1 2 β’ ππ¦ = π£ππ¦ π‘ + β ππ‘ 2 β’ This is the same equation you used before, except that before the initial vertical velocity was zero and dropped out of the equation. Example problem β’ A ball is thrown 30 degrees with an initial velocity of 10.0 m/s. How long will it take to fall? How high did it go? Example problem 2 β’ You throw a ball 45 degrees upwards with an initial velocity of 10.0 m/s. Assuming you are on flat ground, how far (horizontally) away from you will the ball land? Summary for solving projectiles launched at angle β’ Solve for vertical velocity using vi*sin(ΞΈ) and horizontal velocity using vi*cos(ΞΈ) β’ Find time to maximum height using vfy = 0 and acceleration equation below β’ vfy = viy + at β’ Multiply time to maximum height by 2 to get total time. β’ You can use the projectile equation (below) to solve for d, vi, or t if you know two of the variables. β’ d = viy*t + 0.5at2
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