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- Give an example of a trajectory in two or three dimensions caused by independent perpendicular motions.Consider a projectile launched form ground level at an angle of elevation with an initial velocity . The maximum horizontal range is given by xmax=(v 2/0 sin2θ)/g, where g is the acceleration due to gravity. Here is the problem: If a soccer ball is kicked from ground level with an initial velocity of 28 m/sec, what is the smallest positive angle at which the player should kick the ball to reach a teammate 48m down the field? Assume that the ball reaches the teammate at ground level on the fly. Round to the nearest tenth of a degree.A fireworks show is choreographed to have two shells cross paths at a height of 154 feet and explode at an apex of 191 feet under normal weather conditions. If the shells have a launch angle θ = 56° above the horizontal, determine the common launch speed v0 for the shells, the separation distance d between the launch points A and B, and the time t from the launch at which the shells explode.
- William Tell is said to have shot an apple off his son’s head with an arrow. If the arrow was shot with an initial speed of 50.0 m/s and the boy was 26.0 m away, at what launch angle above the horizontal did Bill aim the arrow? (Assume that the arrow and apple are initially at the same height above the ground. Hint: sin2θ = 2sinθcosθ)Miltiadis Tentoglou is the current Olympic gold medalist for the running long jump. In one of Miltiadis’ practices, he insanely attempts to jump five people who are laid out on the ground in single file from head to toe. Each of the five individuals are 1.7 meters in height and when Miltiadis launches off the ground his initial resultant velocity is directed 20 degrees above the horizon and the horizontal component of this resultant velocity is 10 m/s. In executing the jump, Miltiadis’ center of mass drops 0.6 meters vertically from the instant of takeoff to the instant of landing. Calculate Miltiadis’ a) initial and final vertical velocities in m/s, b) initial and final resultant velocity magnitudes in m/s, c) total time in the air in seconds and d) how far he was able to jump horizontally in meters. e) How many centimeters of clearance did Miltiadis have on both sides of the jump when jumping the five individuals (assume the clearance distance is the same on both sides of the…A projectile is launched with speed v at an angle of ?2 above the horizontal going to a maximum height NH=4.08 H.The projectile is then launched with the same initial speed but an angle of ?1 above the horizontal going to a maximum height H.Please derive the ratio of the total time of flight of the projectile as the time for launch at angle ?2 to that time at angle ?1.
- From the window of a building, a ball is tossed from a height y0 above the ground with an initial velocity of 8.60 m/s and angle of 18.0° below the horizontal. It strikes the ground 5.00 s later. (a) If the base of the building is taken to be the origin of the coordinates, with upward the positive y-direction, what are the initial coordinates of the ball? (Use the following as necessary: y0.) xi = 0 yi = γ0 (b) With the positive x-direction chosen to be out the window, find the x- and y-components of the initial velocity. vi,x = m/s vi,y = m/s (c) Find the equations for the x- and y- components of the position as functions of time. (Use the following as necessary: y0 and t. Let the variable t be measured in seconds.) x = m y = m (d) How far horizontally from the base of the building does the ball strike the ground? m(e) Find the height from which the ball was thrown. m(f) How long does it take the ball to reach a point 10.0…Consider two projectiles whose initial position is the same height, y = 5m. Projectile 1 is fired horizontally with initial velocity horizontal speed v = 10 m/s, while Projectile 2 is dropped. How far has Projectile 1 traveled horizontally when it strikes the ground?Two projectiles are thrown with the same initial speed, one at an angle u with respect to the level ground and the other at angle 90° − θ. Both projectiles strike the ground at the same distance from the projection point. Are both projectiles in the air for the same length of time?
- A ball is tossed horizontally from the highest point of a 75 m building and lands 150 m from the base of the building. Ignore air resistance, and use a coordinate system whose origin is at the highest point of the building, with positive y upwards and positive x in the direction of the throw. How long is the ball in the air for, in seconds?A particle initially located at the origin has an acceleration of = 2.00ĵ m/s2 and an initial velocity of i = 7.00î m/s. (a) Find the vector position of the particle at any time t (where t is measured in seconds) (b) Find the velocity of the particle at any time t. (c) Find the coordinates of the particle at t = 8.00 s (d) Find the speed of the particle at t = 8.00 sIn the figure, a ball is thrown leftward from the left edge of the roof, at height h above the ground. The ball hits the ground 2.10 s later, at distance d = 26.0 m from the building and at angle θ = 65.0° with the horizontal. (a) Find h. (Hint: One way is to reverse the motion, as if on videotape.) What are the (b) magnitude and (c) angle relative to the horizontal of the velocity at which the ball is thrown (positive angle for above horizontal, negative for below)?