1. Intro to Physics Units

A box moving with an initial speed v is accelerated horizontally. If x is measured in [m],
v in [m/s], a in [m/s²], t in [s] which of the following equations is correct for solving the distance x?

Newton's Law of Gravitation describes the attraction force between two masses. The equation is ,

where F is in [ kg·m / s²], m₁ and m₂ are masses in [ kg], and r is the distance in [ m] between them.

Determine the units of the Universal Constant G.

How many significant figures do each of the following numbers have:

(e) 0.0086 ?

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.

(b) In shallow water, the speed of surface waves is found experimentally to be independent of the wavelength (i.e., γ = 0 in our assumed equation above for v). Using only dimensional analysis, determine the formula for the speed of waves in shallow water.

(II) (a) Show that the following combination of the three fundamental constants of nature that we used in Example 1–10 (that is G, c, and h) forms a quantity with the dimensions of time:

tₚ = √Gh/c⁵

This quantity, tₚ, is called the Planck time and is thought to be the earliest time, after the creation of the Universe, at which the currently known laws of physics can be applied.

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.

(a) 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.

The following formula estimates an average person's lung capacity V (in liters, where 1 L = 10³ cm³):

V = 4.1H ― 0.018A ―2.7 ,

where H and A are the person's height (in meters) and age (in years), respectively. In this formula, what are the units of the numbers 4.1, 0.018, and 2.7?

(II) The speed v of an object is given by the equation v = At³ ― Bt, where t refers to time.

(b) What are the SI units for the constants A and B?

The position of a ball rolling in a straight line is given by 𝓍 = 2.0 ― 3.6t + 1.7t², where 𝓍 is in meters and t in seconds.

(a) What do the numbers 2.0, 3.6, and 1.7 refer to?