Concept explainers
(a)
The equation for
(a)
Answer to Problem 44AP
The equation for
Explanation of Solution
Given info: The given equation is
The equation for the kinetic energy is given as,
Here,
Rearrange the above equation for
Let us assume
Further solve the equation.
Replace
Conclusion:
Therefore, the equation for
(b)
The minimum possible value of speed and corresponding kinetic energy.
(b)
Answer to Problem 44AP
The minimum possible value of speed can be zero and corresponding kinetic energy will also be zero.
Explanation of Solution
Given info: The given equation is
From equation (2), the expression for the speed is given as,
From the above expression all the term is positive as well as the expression contains only positive sign so the minimum possible value that the speed can have according to the above expression is zero.
At zero speed the corresponding value of kinetic energy is also zero.
Conclusion:
Therefore, the minimum possible value of speed can be zero and corresponding kinetic energy will also be zero.
(c)
The maximum possible value of speed and corresponding kinetic energy.
(c)
Answer to Problem 44AP
The maximum possible value of speed can be speed of light and corresponding kinetic energy will increases without any limit.
Explanation of Solution
Given info: The given equation is
From equation (2), the expression for the speed is given as,
The maximum value of speed is equal to the speed of light according to relativistic concept if the speed becomes more than the speed of light then its energy become unstable that would not exist practically.
At this speed of light, the kinetic energy increases without any limit.
Conclusion:
Therefore, the maximum possible value of speed can be speed of light and corresponding kinetic energy will increases without any limit.
(d)
The equation for the acceleration of the particle as a function of kinetic energy and power input.
(d)
Answer to Problem 44AP
The equation for the acceleration of the particle as a function of kinetic energy and power input is
Explanation of Solution
Given info: The given equation is
From equation (1), the expression for the speed is given as,
Write the expression for the acceleration of a particle.
Substitute
Replace
Substitute
Conclusion:
Therefore, the equation for the acceleration of the particle as a function of kinetic energy and power input is
(e)
The limiting form of the expression in part (d) at low energy and compare with the non-relativistic expression.
(e)
Answer to Problem 44AP
The limiting form of the expression of acceleration at low energy is
Explanation of Solution
Given info: The non-relativistic expression for the acceleration is
From equation (4), the expression for the acceleration is given as,
At low energy the value of
Thus, the limiting form of the expression of acceleration at low energy is
Conclusion:
Therefore, the limiting form of the expression of acceleration at low energy is
(f)
The limiting form of the expression in part (d) at high energy and compare with the non-relativistic expression.
(f)
Answer to Problem 44AP
The limiting form of the expression of acceleration at high energy is
Explanation of Solution
Given info: The non-relativistic expression for the acceleration is
From equation (4), the expression for the acceleration is given as,
At high energy the value of
Thus, the limiting form of the expression of acceleration at low energy is
Conclusion:
Therefore, the limiting form of the expression of acceleration at high energy is
(g)
The reason that answer to part (f) help account for the answer to part (c) at constant input power.
(g)
Answer to Problem 44AP
The acceleration of the particle is very less at high energy that gives the velocity of the particle a constant value.
Explanation of Solution
Given info: The non-relativistic expression for the acceleration is
From the answer of part (f) the expression for the acceleration is,
Here,
In part (c), the speed at high energy approaches to the speed of light. But from the acceleration equation if the energy is imparted to the particle at constant input power the acceleration is steeply decreases because the acceleration is inversely proportional to the cube root of the kinetic energy. So at high energy acceleration is very less and the velocity of the particle approaches to a constant value as indicate in part (c).
Conclusion:
Therefore, the acceleration of the particle is very less at high energy that gives the velocity of the particle a constant value.
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Chapter 38 Solutions
Physics for Scientists and Engineers with Modern Physics
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