Can you please show how to derive θ = 1/2αt2 + ω0t + θ0

Which vector has an angle with respect to the positive x-axis that is in the range of the inverse tangent function?

Show that, when SI units for 0 and 0 are entered, the units given by the right-hand side of the equation in the problem above are m/s.

Obtain the sine law of plane trigonometry by interpreting the product A × B and the alternate representation (A − B) × B.

You stand in a flat meadow and observe two cows (Fig. P1.73). Cow A is due north of you and 15.0 m from your position. Clow B is 25.0111 from your position. From your point of view, the angle between cow A and cow H is 20.0, with cow B appearing to the right of cow A. (a) How far apart are cow A and cow B? (b) Consider the view seen by cow A. According to this cow, what is the angle between you and cow B? (c) Consider the view seen by cow B. According to this cow, what is the angle between you and cow A? Hint: What does the situation look like to a hummingbird hovering above the meadow? (d) Two stars in the sky appear to be 20.0 apart. Star A is 15.0 ly from the Earth, and star B. appearing to the right of star A. is 25.0 ly from the Earth. To an inhabitant of a planet orbiting star A. what is the angle in the sky between star B and our Sun? Figure P1.73 Your view of two cows in a meadow. Cow A U due north of you. You must rotate your eyes through an angle of 20.0 to look from cow A to cow B.

CASE STUDY In Example 34.6 (page 1111), we imagined equipping 1950DA, an asteroid on a collision course with the Earth, with a solar sail in hopes of ejecting it from the solar system. We found that the enormous size required for the solar sail makes the plan impossible at this time. Of course, there is no need to eject such an object from the solar system: we only need to change the orbit. A much more pressing problem is Apophis, a 300-m asteroid that may be on a collision course with the Earth and is due to come by on April 13, 2029. It is unlikely to hit the Earth on that pass, but it will return again in 2036. If Apophis passes through a 600-m keyhole on its 2029 pass, it is expected to hit the Earth in 2036. causing great damage. There are plans to deflect Apophis when it comes by in 2029. For example, we could hit it with a 10- to 150-kg impactor accelerated by a solar sail. The impactor is launched from the Earth to start orbiting the Sun in the same direction as the Earth and Apophis. The idea is to use a solar sail to accelerate the impactor so that it reverses direction and collides head-on with Apophis at 8090 km/s and thereby keeps Apophis out of the keyhole. Consider the momentum in the impactors orbit (Fig. P34.75) when the solar sail makes an angle of = 60 with the tangent to its orbit. Current solar sails may be about 40 m on a side, but the hope is to construct some that are about 160 m on a side. Estimate the impactors tangential acceleration when it is about 1 AU from the Sun. Keep in mind that the sail is neither a perfect absorber nor a perfect reflector, and a heavier impactor would presumably be equipped with a larger sail. Dont be surprised by what may seem like a very small acceleration. FIGURE P34.75

Figure P1.10 shows a frustum of a cone. Match each of the expressions (a) (r1 + r2)[h2 + (r2 r,)2] (b) 2(r1 + r2), and (c) h(r12 + r1r2 + r22)/3 (d) the total circumference of' the flat circular faces, (e) the volume, or (F) the area of the curved surface.

A farmer wants to fence off his four-sided plot of flat land. He measures the first three sides, shown as A, B, and C in Figure 3.62, and then correctly calculates the length and orientation of the fourth side D. What is his result?

Describe the following physical occurrences as events, that is, in the form (x, y, z, t): (a) A postman rings a doorbell of a house precisely at noon. (b) At the same lime as the doorbell is lung, a slice of bread pops out of a toaster that is located 10 1T1 from the door in the east direction from the door. (c) Tell seconds later, an airplane arrives at the airport, which is 10 km from the door in the east direction and 2 km to the south.

You are working in your dream job: an assistant for the special effects department of a movie studio. You have just been given this assignment: the star of a horror movie is walking down a spooky hallway when suddenly, due to some unknown and strange supernatural forces, all the pictures hanging on the wall start rotating about their upper edges until they are sticking straight out from the wall! To set up this effect, you attach the pictures to the wall with hinges along their upper end and wrap 20 turns of wire around the outside frame of the picture, as shown in Figure P28.32a. You set up a uniform magnetic field in the hallway that is directed upward and oriented at an angle of = 5.00 to the vertical, with its horizontal component directed perpendicularly into the wall. When you send a current of I = 10.0 A through the wire around each picture, the frame swings up perpendicular to the wall as shown in Figure P28.32b. Consider a particular picture of width = 40.6 cm, height h = 50.8 cm, and mass m = 0.750 kg. (a) Your supervisor asks you to determine the magnetic field magnitude that is necessary for this picture to rotate so that its face is parallel to the floor and perpendicular to the wall, as in Figure P28.32b. (b) She also asks about any dangers associated with this magnetic field.