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Chapter 1: Problem 53

Suppose quantity \(s\) is a length and quantity \(t\) is a time. Suppose the quantities \(v\) and \(a\) are defined by \(v\) \(=d s / d t\) and \(a=d v / d t .\) (a) What is the dimension of \(v ?\) (b) What is the dimension of the quantity \(a ?\) What are the dimensions of (c) \(\int v d t\) (d) \(\int a d t,\) and (e) \(d a / d t ?\).

Short Answer

Expert verified
The dimensions of the given quantities are: (a) \(v = [LT^{-1}]\), (b) \(a = [LT^{-2}]\), (c) \(\int v dt = [L]\), (d) \(\int a dt = [LT^{-1}]\), and (e) \(\frac{d a}{d t} = [LT^{-3}]\).

Step by step solution

01

Find the dimensions of \(v\)

Given that \(v = \frac {d s}{d t}\) and s is a length, we have the following dimensions: \[v = \frac{[L]}{[T]}\] where [L] is the dimension of length and [T] is the dimension of time. So the dimensions of \(v\) are: \[v = [LT^{-1}]\]
02

Find the dimensions of \(a\)

Given that \(a = \frac{d v}{d t}\) and from the previous step, we know that the dimensions of \(v\) are [\(LT^{-1}\)], we can find the dimensions of \(a\) as: \[a = \frac{[LT^{-1}]}{[T]}\] So the dimensions of \(a\) are: \[a = [LT^{-2}]\]
03

Find the dimensions of \(\int v dt\)

To find the dimensions of \(\int v dt\), we integrate the dimensions of \(v\) with respect to time: \[\int [LT^{-1}] dt\] Since we are integrating with respect to time, the dimensions become: \[[LT^{-1}] [T] = [L]\] So the dimensions of \(\int v dt\) are [L] (length).
04

Find the dimensions of \(\int a dt\)

To find the dimensions of \(\int a dt\), we integrate the dimensions of \(a\) with respect to time: \[\int [LT^{-2}] dt\] Since we are integrating with respect to time, the dimensions become: \[[LT^{-2}] [T] = [LT^{-1}]\] So the dimensions of \(\int a dt\) are [\(LT^{-1}\)].
05

Find the dimensions of \(\frac{d a}{d t}\)

To find the dimensions of \(\frac{d a}{d t}\), we differentiate the dimensions of \(a\) with respect to time: \[\frac{d [LT^{-2}]}{d t}\] Since we are differentiating with respect to time, the dimensions become: \[[LT^{-2}] [T^{-1}] = [LT^{-3}]\] So the dimensions of \(\frac{d a}{d t}\) are [\(LT^{-3}\)].

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Kinematics
Kinematics is a branch of physics that focuses on the motion of objects without considering the causes of this motion, such as forces or energy. It primarily deals with quantities like displacement, velocity, and acceleration. In kinematics, we often describe the position of a body in terms of its path or trajectory.

This field uses various mathematical equations and graphical representations to describe how an object moves. Key variables in kinematics include:
  • Displacement: The change in position of an object. It is a vector quantity, which means it has both direction and magnitude.
  • Velocity: This refers to the rate of change of displacement. Like displacement, it is a vector quantity.
  • Acceleration: The rate at which an object's velocity changes over time.
Understanding these concepts is fundamental for solving problems related to motion in physics. By grasping kinematics, students can better analyze and predict the behavior of moving objects.
Velocity
Velocity is a key concept in kinematics and is defined as the rate of change of displacement with respect to time. It is distinct from speed, which is only concerned with how fast an object is moving, as opposed to its direction.

Mathematically, velocity is expressed as:
  • \[ v = \frac{ds}{dt} \]
  • where \( v \) is velocity, \( s \) is displacement, and \( t \) is time.
The dimension of velocity is [LT extsuperscript{-1}], which combines length (L) and time (T), emphasizing that velocity quantifies how fast and in what direction an object is moving. A constant velocity indicates uniform motion, whereas changing velocity suggests acceleration.
Acceleration
Acceleration describes how the velocity of an object changes over time. It is a vector quantity, meaning it has both magnitude and direction. Acceleration can occur as an increase in speed, a decrease (deceleration), or a change in direction.

The mathematical representation of acceleration is:
  • \[ a = \frac{dv}{dt} \]
  • where \( a \) is acceleration and \( v \) is velocity.
The dimensions of acceleration are expressed as [LT extsuperscript{-2}]. This reflects a change in velocity over time, indicating that acceleration involves how quickly an object speeds up, slows down, or alters its direction. Exploring acceleration helps us better understand varying motions experienced in real-world scenarios.
Integration in Physics
Integration is a fundamental mathematical process used in physics to find quantities like displacement or velocity from their rates of change. It is often necessary when dealing with differential equations or determining an area under a curve, such as a velocity-time graph.

For instance, integrating velocity gives displacement:
  • \[ \int v \, dt = s \]
  • Here, \( s \) represents displacement.
Similarly, integrating acceleration grants velocity:
  • \[ \int a \, dt = v \]
  • which shows how we can retrieve velocity from acceleration over time.
These integrations highlight how understanding the relationships between physical quantities in physics requires adept manipulation of calculus concepts to describe motion comprehensively.

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Most popular questions from this chapter

Determine the number of significant figures in the following measurements: (a) 0.0009 ,(b) 15,450.0, (c) \(6 \times 10^{3}\), (d) 87.990, (e) 30.42

The first atomic bomb was detonated on July 16,1945 at the Trinity test site about \(200 \mathrm{mi}\) south of Los Alamos. In 1947 , the U.S. government declassified a film reel of the explosion. From this film reel, British physicist G. I. Taylor was able to determine the rate at which the radius of the fireball from the blast grew. Using dimensional analysis, he was then able to deduce the amount of energy released in the explosion, which was a closely guarded secret at the time. Because of this, Taylor did not publish his results until 1950. This problem challenges you to recreate this famous calculation. (a) Using keen physical insight developed from years of experience, Taylor decided the radius \(r\) of the fireball should depend only on time since the explosion, \(t,\) the density of the air, \(\rho,\) and the energy of the initial explosion, \(E .\) Thus, he made the educated guess that \(r=k E^{a} \rho^{b} t^{c}\) for some dimensionless constant \(k\) and some unknown exponents \(a, b,\) and \(c .\) Given that \([\mathrm{E}]=\mathrm{ML}^{2} \mathrm{T}^{-2}\) determine the values of the exponents necessary to make this equation dimensionally consistent. (Hint: Notice the equation implies that \(k=r E^{-a} \rho^{-b} t^{-c}\) and that \([k]=1\) ) (b) By analyzing data from high-energy conventional explosives, Taylor found the formula he derived seemed to be valid as long as the constant \(k\) had the value \(1.03 .\) From the film reel, he was able to determine many values of \(r\) and the corresponding values of \(t .\) For example, he found that after \(25.0 \mathrm{ms}\), the fireball had a radius of \(130.0 \mathrm{m}\). Use these values, along with an average air density of 1.25 \(\mathrm{kg} / \mathrm{m}^{3},\) to calculate the initial energy release of the Trinity detonation in joules (J). (Hint: To get energy in joules, you need to make sure all the numbers you substitute in are expressed in terms of SI base units.) (c) The energy released in large explosions is often cited in units of "tons of TNT" (abbreviated "t TNT"), where 1 t TNT is about 4.2 GJ. Convert your answer to (b) into kilotons of TNT (that is, kt TNT). Compare your answer with the quickand-dirty estimate of \(10 \mathrm{kt}\) TNT made by physicist Enrico Fermi shortly after witnessing the explosion from what was thought to be a safe distance. (Reportedly, Fermi made his estimate by dropping some shredded bits of paper right before the remnants of the shock wave hit him and looked to see how far they were carried by it.)

Find the order of magnitude of the following physical quantities. (a) The mass of Earth's atmosphere: \(5.1 \times 10^{18} \mathrm{kg} ;\) (b) The mass of the Moon's atmosphere: \(25,000 \quad \mathrm{kg} ; \quad\) (c) The mass of Earth's hydrosphere: \(1.4 \times 10^{21} \mathrm{kg} ;\) (d) The mass of Earth: \(5.97 \times 10^{24} \mathrm{kg}\) (e) The mass of the Moon: \(7.34 \times 10^{22} \mathrm{kg} ;\) (f) The Earth-Moon distance (semimajor axis): \(3.84 \times 10^{8} \mathrm{m} ;(\mathrm{g})\) The mean Earth-Sun distance: \(1.5 \times 10^{11} \mathrm{m} ;\) (h) The equatorial radius of Earth: \(6.38 \times 10^{6} \mathrm{m} ;\) (i) The mass of an electron: \(9.11 \times 10^{-31} \mathrm{kg} ;\) (j) The mass of a proton: \(1.67 \times 10^{-27} \mathrm{kg} ; \quad\) (k) The mass of the Sun: \(1.99 \times 10^{30} \mathrm{kg}\).

The arc length formula says the length \(s\) of arc subtended by angle \(\sim\) in a circle of radius \(r\) is given by the equation \(s=r \Theta .\) What are the dimensions of (a) \(s\) (b) \(r,\) and \((\mathrm{c}) \Theta ?\)

What determines the validity of a theory?
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