Physics 410
Spring 2014
HW#4
Due Friday, 14 February 2014
1. Consider F(v) = A (a constant) for vo/2 ≤ vo ≤ 3vo/2 and = 0 otherwise.
a) Sketch a plot of F(v) vs. v.
b) Normalize this to find A in terms of vo.
c) Determine expressions for vavg and vrms in terms of vo using this F(v).
2. Carter, 11.1 but skip part d: The distribution of particle speeds of a certain
hypothetical gas is given by
𝑁(𝑣) 𝑑𝑣 = 𝐴𝑣𝑒 −𝑣/𝑣𝑜 𝑑𝑣
where 𝐴 and vo are constants.
𝑁(𝑣)
a) Determine 𝐴 so that 𝑓(𝑣) = 𝑁 is a true probability density function; i.e.,
∞
∫0 𝑓(𝑣)𝑑𝑣 = 1. Sketch 𝑓(𝑣) versus 𝑣.
b) Find 𝑣̅ and 𝑣𝑟𝑚𝑠 in terms of 𝑣𝑜 .
c) Differentiate 𝑓(𝑣) with respect to 𝑣 and set the result equal to zero to find the most
probable speed 𝑣𝑚 .
3. Carter, 11.8: Compute 𝑣𝑚 , 𝑣̅ , and 𝑣𝑟𝑚𝑠 for an oxygen molecule at 300K. What are
the corresponding values at 10,000K?
Physics 410
Spring 2014
HW#4
Due Friday, 14 February 2014
1. Consider F(v) = A (a constant) for vo/2 ≤ vo ≤ 3vo/2 and = 0 otherwise.
a) Sketch a plot of F(v) vs. v.
b) Normalize this to find A in terms of vo.
c) Determine expressions for vavg and vrms in terms of vo using this F(v).
2. Carter, 11.1 but skip part d: The distribution of particle speeds of a certain
hypothetical gas is given by
𝑁(𝑣) 𝑑𝑣 = 𝐴𝑣𝑒 −𝑣/𝑣𝑜 𝑑𝑣
where A and vo are constants.
𝑁(𝑣)
a) Determine 𝐴so that 𝑓(𝑣) = 𝑁 is a true probability density function; i.e.,
∞
∫0 𝑓(𝑣)𝑑𝑣 = 1. Sketch 𝑓(𝑣) versus 𝑣.
b) Find 𝑣̅ and 𝑣𝑟𝑚𝑠 in terms of 𝑣𝑜 .
c) Differentiate 𝑓(𝑣) with respect to 𝑣 and set the result equal to zero to find the
most probable speed 𝑣𝑚 .
3. Carter, 11.8: Compute 𝑣𝑚 , 𝑣̅ , and 𝑣𝑟𝑚𝑠 for an oxygen molecule at 300K. What are
the corresponding values at 10,000K?