1. ( 26 points) This problem deals with a building in a flooded region of a tropical country. A ball is thrown from the top of the building whose roof is 50.0 m above the surface of a lake created by heavy rainfall. The initial speed is |v1| =20.0 m/s. The ball follows the one-dimensional trajectory schematically shown.
Assume the magnitude of gravitational acceleration is g = 9.80 m/s2.
The ball hits the water surface and moves downward to the bottom of the lake with a downward acceleration whose magnitude is |a | = 3.00 m/s2. Just before it hits the bottom of the lake, the speed is |v5| = 40.0 m/s. (Scales shown in the schematic diagram below are not necessarily accurate to the parameters of the problem.)
(a) (2 points) What is the speed |v2 | at point 2 at the top of the trajectory ?
(b) (3 points) What is the magnitude |a | of the acceleration at point 2 at the top of the trajectory ?
(c) (5 points) What is the time that it takes to reach point 2 at the top of the trajectory ?
(d) (2 points) What is the speed |v3| when the ball returns to the level of the roof ?
(e) (14 points) What is the depth D of the lake ?
| 1. Answers (a) v2 = 0. (b) a = 9.8 m/s2 (surprise!) (c) t = 2.04 sec. Use 0 = 20 - gt (d) v3 = 20 - g*4 = -20 m/s (e) v42 = v32 + 2g*50 so that v4 =-37.1 m/s v52 = v42 + 2a*(delta y) a = -3m/s2 so that delta y = -36.77 m. Thus depth = 36.77 m |
2. ( 20 points)
A bullet is fired horizontally into a wooden block that is rigidly attached to a table whose top surface is a height h = 1.0 m above the ground. The bullet moves horizontally just above where the bottom of the block and the table top are in contact. The speed of the bullet just before it enters the wooden board is |V1| =
420 m/s. Assume the acceleration of the bullet while in the block is horizontal and negative and has the value a = - 6.1x105 m/s2. We assume that the block is 10.0 cm thick. (Note: 1.0 cm = 0.010 m.) The bullet emerges from the right side of the block with speed |V2|. (The scales in the schematic diagram below are not necessarily accurate to the parameters of the problem.)
The bullet continues forward in a projectile motion type trajectory and hits the ground a horizontal distance D from the right end of the table.
(In this problem, ignore the length of the bullet. The bullet is assumed to be so short that there are no "transition" effects as it enters and leaves the wooden block.)
(a) 7 points)How long (in seconds) does it take the bullet to hit the ground after it passes through the wooden block?
(b) (13 points) What is the horizontal distance D from the right end of the table?
| 2. Answers (a) 1 = ½gt2 t = 0.45 sec. (b) v22 = v12 -2(610000)(0.1) v2 = 233 m/s D = 233*0.45 = 105 m |
3. ( 20 points) The 3 vectors are oriented as shown in the figure below.
Both the vector-A and the vector-B have related angles of 45 degrees with the x-axis.
Find:
(a) (7 points) the x and y components Rx and Ry of the resultant vector given by:
(b) (6 points) the magnitude of 
(c) (7 points) The direction of the resultant vector. What is the related angle with the x-axis? In what quadrant does the resultant vector point? Make a sketch of the resultant on the axes provided. Show the related angle in the sketch.
| 3. Answers Rx = -8cos45 + 40cos45 = 22.62 Ry = -8sin45 - 40sin45 + 10 = -23.94 (b) R2 = 22.622 + (-23.94)2 Thus, R = 32.94 (c) tan Ø = 23.94/22.62 so Ø = 460. |
4. Extra Credit Fun! The two blocks below are connected by a string that is wrapped around a pulley at the top of the inclined plane shown. The masses are given by m1 = 20.0 kg and m2 = 2.0 kg . The coefficient of kinetic friction between the block m2 and the inclined plane is 0.10. The system is accelerating and speeding up.
(a) (4 points) What is the magnitude of the acceleration a of the blocks?
(b) (3 points) What is the tension T in the string?
| 4. Answers (1)a = T - (1)g sin60 - (0.1)g cos60 (20)a = (20)g - T . |