Physics: An Introduction Chapter 3 -- Selected Solutions

Chapter 3: Vectors in Physics
Selected Solutions

Selected Solutions

15. (a)

(b) From the information given in problem 14, we can determine the vectors A and B by calculating the numerical value of their components.

Now that we know the components of D we can find its magnitude D and its direction q:

19. From the given information we can write vector A as , and vector B as

(b)

The direction is below the negative x-axis. Therefore, q = 180^o + 26.6^o = 207^o. This answer is also equivalent to -153^o.

(c)

23. (a) Using the component method the direction and magnitude of A are

(b) The direction and magnitude of B are , or 68.2 degrees above the -y axis. This result is equivalent to q = 180^o - 68.2^o = 110^o to 2 significant figures relative to the +x axis.

(c)

33. (a)

(b) The components will be halved.

41. The solution will be easier to follow if we make the following definitions:

v_pg = velocity of the plane relative to the ground
v_pa = velocity of the plane relative to the air
v_ag = velocity of the air relative to the ground

(a) We are interested in the direction of the plane should be headed which is different from the direction in which it will actually travel. The reason for this difference is because of the air and so we need the direction of v_pa. A velocity addition involving v_pa is

v_pg = v_pa + v_ag

The magnitude of v_pa is given to be 310 km/h, and so we can write the velocity vector as

where q is the direction we seek to determine and take east to be along the positive x-axis and north to be along the positive y-axis.

Since east is along the positive x-axis we also have . The velocity addition equation then becomes

For the plane to travel due north, the net velocity in the x-direction relative to the ground must equal zero.

(b)

(c) If the plane's speed is decreased from 310 km/h then the factor cos q will need to increase so that the product of the two will equal 75 km/h. The factor cos q will increse if q increases.