1. Intro to Physics Units

Dimensional analysis. Waves on the surface of the ocean do not depend significantly on the properties of water such as density or surface tension. The primary 'return force' for water piled up in the wave crests is due to the gravitational attraction of the Earth. Thus the speed v (m/s) of ocean waves depends on the acceleration due to gravity g. It is reasonable to expect that υ might also depend on water depth h and the wave's wavelength λ. Assume the wave speed is given by the functional form v = Cgᵅ hᵝ λᵞ, where α , β , c and C are numbers without dimension. In deep water, the water deep below the surface does not affect the motion of waves at the surface. Thus υ should be independent of depth h (i.e., β = 0). Using only dimensional analysis (Section 1–7 and Appendix D), determine the formula for the speed of surface ocean waves in deep water.

(II) Estimate how many books can be shelved in a college library with 6500 m² of floor space. Assume 8 shelves high, having books on both sides, with corridors 1.5 m wide. Assume books are about the size of this one, on average.

(II) Estimate how long it would take one person to mow a football field using an ordinary home lawn mower (Fig. 1–12). (State your assumptions, such as the mower moves with a 1-km/h speed, and has a 0.5-m width.) <IMAGE>

(II) A hiking trail is 270 km long through varying terrain. A group of hikers cover the first 49 km in two and a half days. Estimate how much time they should allow for the rest of the trip.

(II) Estimate the number of jelly beans in the jar of Fig. 1–13.

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Recent findings in astrophysics suggest that the observable universe can be modeled as a sphere of radius R = 13.7 x 10⁹ light-years = 13.0 x 10²⁵ m with an average total mass density of about 1 x 10⁻^-26 kg/m³. Only about 4% of total mass is due to 'ordinary' matter (such as protons, neutrons, and electrons). Estimate how much ordinary matter (in kg) there is in the observable universe. (For the light-year, see Problem 25.)

(II) The diameter of the planet Mercury is 4879 km. What is the surface area of Mercury?

One mole of atoms consists of 6.02 x 10²³ individual atoms. If a mole of atoms were spread uniformly over the Earth's surface, how many atoms would there be per square meter?

What are the rest energy, the kinetic energy, and the total energy of a 1.0 g particle with a speed of 0.80c?

Relativistic Baseball. Calculate the magnitude of the force required to give a 0.145 kg baseball an acceleration a = 1.00 m/s² in the direction of the baseball's initial velocity when this velocity has a magnitude of (a) 10.0 m/s; (c) 0.990c.

A proton (rest mass $1.67\times {10}^{-27}$ kg) has total energy that is $4.00$ times its rest energy. What are (a) the kinetic energy of the proton; (b) the magnitude of the momentum of the proton; (c) the speed of the proton?

Electrons are accelerated through a potential difference of $750$ kV, so that their kinetic energy is $7.50\times {10}^5$ eV.

(a) What is the ratio of the speed $v$ of an electron having this energy to the speed of light, $c$?

(b) What would the speed be if it were computed from the principles of classical mechanics?

Compute the kinetic energy of a proton (mass $1.67\times {10}^{-27}$ kg) using both the nonrelativistic and relativistic expressions, and compute the ratio of the two results (relativistic divided by nonrelativistic) for speeds of (a) $8.00\times {10}^7$ m/s and (b) $2.85\times {10}^8$ m/s.

Why Are We Bombarded by Muons? Muons are unstable subatomic particles that decay to electrons with a mean lifetime of 2.2 μs. They are produced when cosmic rays bombard the upper atmosphere about 10 km above the earth's surface, and they travel very close to the speed of light. The problem we want to address is why we see any of them at the earth's surface. (a) What is the greatest distance a muon could travel during its 2.2 μs lifetime? (b) According to your answer in part (a), it would seem that muons could never make it to the ground. But the 2.2 μs lifetime is measured in the frame of the muon, and muons are moving very fast. At a speed of 0.999c, what is the mean lifetime of a muon as measured by an observer at rest on the earth? How far would the muon travel in this time? Does this result explain why we find muons in cosmic rays? (c) From the point of view of the muon, it still lives for only 2.2 μs, so how does it make it to the ground? What is the thickness of the 10 km of atmosphere through which the muon must travel, as measured by the muon? Is it now clear how the muon is able to reach the ground?

As you pilot your space utility vehicle at a constant speed toward the moon, a race pilot flies past you in her spaceracer at a constant speed of $0.800c$ relative to you. At the instant the spaceracer passes you, both of you start timers at zero.

(a) At the instant when you measure that the spaceracer has traveled $1.20\times {10}^8$ m past you, what does the race pilot read on her timer?

(b) When the race pilot reads the value calculated in part (a) on her timer, what does she measure to be your distance from her?

(c) At the instant when the race pilot reads the value calculated in part (a) on her timer, what do you read on yours?