5.1 m is therefore the maximum height of the ball above the window. This is 25.1 m above the ground. (b)

Transcription

1 .6. Model: This is a case of free fall, so the su of the kinetic and gravitational potential energy does not change as the ball rises and falls. The figure shows a ball s before-and-after pictorial representation for the three situations in parts (a, (b and (c. Solve: The quantity K + U g is the sae during free fall: Kf + Ugf = Ki + Ugi. We have (a v + gy = v + gy y = v v g [( s / ( s / ]/( 9. 8s / 5. ( = = 5. is therefore the axiu height of the ball above the window. This is 5. above the ground. (b v + gy = v + gy Since y = y =, we get for the agnitudes v = v = / s. (c v3 + gy3 = v + gy v3 + gy3 = v + gy v3 = v + g( y y3 v 3 = ( / s + ( 9. 8 / s [ ( ] = 49 /s This eans the agnitude of v 3 is equal to. /s. Assess: Note that the ball s speed as it passes the window on its way down is the sae as the speed with which it was tossed up, but in the opposite direction.

2 .. Model: Model the ball as a particle undergoing rolling otion with zero rolling friction. The su of the ball s kinetic and gravitational potential energy, therefore, does not change during the rolling otion. Solve: Since the quantity K + U g does not change during rolling otion, the energy conservation equations apply. For the linear segent the energy conservation equation K + U = K + U is g g v + gy = v + gy + = v g( ( s / + gy v = gy For the parabolic part of the track, K + U = K + U is g g v + gy = v + gy v + g( = ( / s + gy v = gy Since fro the linear segent we have v = gy, we get gy = gy or y = y =.. Thus, the ball rolls up to exactly the sae height as it started fro. Assess: Note that this result is independent of the shape of the path followed by the ball, provided there is no rolling friction. This result is an iportant consequence of energy conservation.

3 .6. Model: This is the case of a perfectly inelastic collision. Moentu is conserved because no external force acts on the syste (clay + block. We also represent our syste as a particle. Solve: (a The conservation of oentu equation p fx = p ix is Using ( v = v and ( v =, we get ix ix v ( + v = ( v + ( v fx ix ix v 5. kg = ( = + kg +.5 kg v = 476 v = 476 ( (. (. v fx ix ix ix (b The initial and final kinetic energies are given by Ki = vi x + vi x = kg v + kg ( /s = kg v 5 5 ( ( (. ( (. Kf = ( + vf x = ( kg +.5 kg (. 476 v =. 9 v K i Kf The percent of energy lost = =. 9 % K % = 95. %. 5 i

4 .4. Model: Model the two packages as particles. Moentu is conserved in both inelastic and elastic collisions. Kinetic energy is conserved only in an elastic collision. Solve: For a package with ass the conservation of energy equation is K + Ug = K + Ug ( v + gy = ( v + gy Using ( v = / s and y =, ( v = gy ( v = gy = 98 (. s / ( 3. = s / (a For the perfectly inelastic collision the conservation of oentu equation is Using ( v = / s, we get p = p ( + ( v = ( v + ( ( v fx ix 3 ( v 3 = ( v / 3= 56. s / (b For the elastic collision, the ass package rebounds with velocity ( v = (.. v 3 = ( /s = 56 /s + 3 The negative sign with (v 3 shows that the package with ass rebounds and goes to the position y 4. We can deterine y 4 by applying the conservation of energy equation as follows. For a package of ass : Kf + Ugf = Ki + Ugi ( v4 + gy4 = ( v3 + gy3 Using ( v = 55. / s, y =, and ( v = /s, we get gy4 = ( 56. s / y4 = 333. c

5 .44. Model: Assue an ideal spring that obeys Hooke s law. Since this is a free fall proble, the echanical energy K + U + U is conserved. Also, odel the safe as a particle. g s We have chosen to place the origin of our coordinate syste at the free end of the spring which is neither stretched nor copressed. The safe gains kinetic energy as it falls. The energy is then converted into elastic potential energy as the safe copresses the spring. The only two forces are gravity and the spring force, which are both conservative, so energy is conserved throughout the process. This eans that the initial energy as the safe is released equals the final energy when the safe is at rest and the spring is fully copressed. Solve: The conservation of energy equation K + U + U = K + U + U g s g s is v + g( y ye + k( y ye = v + g( y ye + k( ye ye Using v = v = /s and y e =, the above equation siplifies to gy + ky = gy g( y y ( kg( 9. 8 /s (. (. 5 k = = = 96. y ( 5. Assess: By equating energy at these two points, we do not need to find how fast the safe was oving when it hit the spring. 5 N/.