A particle has a position vector given by r = 36t i + (45t - 9t²) j, where r is in meters and t in seconds. Find the instantaneous-velocity and instantaneous-acceleration vectors as functions of time t. (Answer using i, j and t, as necessary.)
instantaneous velocity

Enter a mathematical expression.

1 m/s
instantaneous acceleration

Enter a mathematical expression.

2 m/s²

A 50 kg woman stands on the back of a 6 m long, 130 kg raft that is floating at rest in still water with no friction. The raft is 0.5 m from a fixed pier, as in Figure 8-63.

Figure 8-63

(a) The woman walks to the front of the raft and stops. How far is the raft from the pier now?

Enter a number.

1 m
(b) While the woman walks, she maintains a constant speed of 3 m/s relative to the raft. Find the total kinetic energy of the system (woman plus raft)

Enter a number.

2 J
Find the total kinetic energy of the system if the woman walked at 3 m/s on a raft tied to the pier. (Make a mental comparison of these two values.)

Enter a number.

3 J
(c) Where does this energy come from, and where does it go when the woman stops at the front of the raft? (Select all that apply.)

4

comes from chemical energy of the woman comes from internal energy of the raft comes from gravity comes from buoyancy goes to internal energy of the raft goes to internal energy of the woman goes to internal energy of the water

(d) On land, the woman puts a lead shot 6 m. She stands at the back of the raft, aims forward, and puts the shot so that just after it leaves her hand, it has the same velocity relative to her as it does when she throws it from the ground. Where does the shot land?

5

The shot lands on the raft and the raft shifts to the right. The shot lands in the water and the raft shifts to the left.     The shot lands on the raft and the raft shifts to the left. The shot lands in the water and the raft shifts to the right.

A spacecraft of 150 kg mass is in a circular orbit about the earth at a height h = 3R_E.

(a) What is the period of the spacecraft's orbit about the earth?

Enter a number.

1 h
(b) What is the spacecraft's kinetic energy?

Enter a number.

2 J
(c) Express the angular momentum L of the spacecraft about the center of the earth in terms of its kinetic energy K. (Answer using R_E for the radius of the earth, K for the Kinetic Energy of the satellite, and m for the mass of the satellite, as needed.)

Enter a mathematical expression.

3
(d) Find the numerical value of the angular momentum.

Enter a number.

4 J · s

Five equal charges Q are equally spaced on a semicircle of radius R as shown in Figure 21-36. Find the force on a charge q located at the center of the semicircle. (Use Q, R, q, k, and the unit vectors i, and j, as necessary.)

Enter a mathematical expression.

1

Figure 21-36

In Figure 27-49, one current is 4 A into the paper, the other current is 4 A out of the paper, and each curve is a circular path.

Figure 27-49

(a) Find _CB · d for each path indicated.
_C1B · d =

Enter a number.

1 T · m
_C2B · d =

Enter an exact number.

2 T · m
_C3B · d =

Enter a number.

3 T · m

(b) Which paths, if any, can be used to find B at some point due to these currents? (Select all that apply.)

4

Path C1 Path C2 Path C3 none of these