Consider the nail-firing device of Example 4.15. When the

QUESTION:

Consider the nail-firing device of Example 4.15. When the device works, the nail is fired with

velocity, 𝑉, with density

                                   \(f(v)=\frac{v^{3} e^{-v / 500}}{(500)^{4} \Gamma(4)}\).

The device misfires 2% of the time it is used, resulting in a velocity of 0. Find the expected kinetic energy associated with a nail of mass 𝑚. Recall that the kinetic energy, 𝑘, of a mass 𝑚

moving at velocity 𝑣 is \(k=\left(m v^{2}\right) / 2\).

The kinetic energy 𝑘 associated with a mass m moving at velocity 𝑣 is given by the expression

                                   \(k=\frac{m v^{2}}{2}\).

Consider a device that fires a serrated nail into concrete at a mean velocity of 2000 feet per second, where the random velocity 𝑉 possesses a density function given by

                                   \(f(v)=\frac{v^{3} e^{-v / 500}}{(500)^{4} \Gamma(4)}, \quad v \geq 0.\)

Find the expected kinetic energy associated with a nail of mass 𝑚.

                                   \(E(K)=E\left(\frac{m V^{2}}{2}\right)=\frac{m}{2} E\left(V^{2}\right),\)

Equation Transcription:

Text Transcription:

f(v)=v^3e^-v/500 over (500)^4 Gamma(4)

k=(mv^2)/2

k=mv^2 over 2

f(v)=v^3e^-v/500 over (500)^4 Gamma(4), v>/=0

E(K)=E(mV^2 over 2)=m over 2 E(V^2)