53. An archer attaches an arrow to a related linear spring of negligible mass, i

Question

53. An archer attaches an arrow to a related linear spring of negligible mass, its relaxed length 12.0 meters, and its free constant 15.0 N/m. The other end of the spring is firmly attached to the ground. The mass of the arrow is 447 grams. The archer shoots the arrow vertically upward, the arrow leaving the bow with a speed V at a point that is 2.00 meters above the ground The arrow stretches the spring to twice its unstretched length before stopping instantaneously. Forty percent of the amow's initial kinetic nergy is dissipated by air friction during the arrow's ascent, calculate v 54, spherical balls of varying mass are to be dropped from rest from the top of the world's tallest building (Burj Khalifa in Dubai). The balls range from 250 grams to 5.00 kilograms in regular increments of 0.250 kg. All of the balls (which are, of course made of different materials) are of equal diameters, D-12.0 centimeters. Use Excel, and create a graph of the rate of energy dissipation at terminal velocity as a function of ball mass. 55. The diagram shows a flat circular slab of negligible thickness that has a square hole cut into it. One side of the hole is along a diameter and centered on the center of the circle. The radius of the circle is R, aed the side length of the square hole is S 4/s R The material of the slab has uniforms density, Calculate, in terms of R, the distance between the center of the circle and the center of mass of the slab. 56, The diagram shows a juggler with three identical balls in the air. One ball, of mass m, is moving horizontally at 20 0 em/s, one ball, of mass 2m, is moving upward with a velocity of 40.0 cm/s at an angle of 12.0° with the vertical, and the third ball, of mass 3m is moving downward with a velocity of 30.0 cm/s at an angle of 14,0° with the vertical. m a 0.250 kg. Calculate: a. The magnitude and direction of the linear momentum of the three-ball system. b. The magnitude and direction of the center of mass of the three-ball system. c. The magnitude and direction of the acceleration of the center of mass of the three-ball system. 57, Three objects, each of mass M, are sliding on a horizontal frictionless surface. One object has a velocity of 4.00 ms due east, the second object has a velocity of 3.00 m's due north. The total linear momentum of the three-object system is zero, M-3.00 kg. Calculate a. The magnitude and direction of the velocity of the third object. b. The total kinetic energy of the three-object system. 58 A ball of mass 255 grams is initially moving horizontally south at 35.0 m s when it encounters a bat. The encounter lasts for only 0.025 seconds immediately after the encounter the ball is moving northward at an angle of 25 below the horizontal with a speed of 45.0 m s. Calculate the magnitude and direction of the average force that the ball exerted on the hat. 59 Bullets are fired at a vertical immovable walL The bullets move horizontally both before and after they strike the wall and rebound with undiminished speed. Each bullet has a mass of 25.0 grams and a speed of 75.0 m/sjust before and after striking the wall. Calculate the rate (in bullets per second) at which the bullets must strike the wall if the wall is to experience an average force of 125 N due the bullets striking it.

Explanation / Answer

53:

mass of the arrow = m = 0.447 kg

initial KE = 0.5 mV2 = 0.2235 V2

The total KE of the arrow is spent in stretching the sprnig , to over come air resitance and reaching a height (24-2) m.

The intial height is 2 m

max height reached = 2*12 -2 = 22 m

sprign stretched by 12 m

PE of the spring Us = 0.5kx2 = 0.5*15*122 = 1080 J

air resistance 40% of initial KE Ur= 0.4*0.2235 V2 = 0.0894 V2

PE of the arrow at max height Uh= mgh = 0.447*9.8*22 = 96.37 J

conserving the total energy

0.2235V2 = Us + Ur + Uh = 1080 +0.0894V2 +96.37

V = 93.66 m/s

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