Answer
(a) $v = 2.8\times 10^8~m/s$
(b) $v = 7.3\times 10^5~m/s$
(c) $v = 730~m/s$
(d) $v = 0.73~m/s$
Work Step by Step
(a) We can find the speed:
$\lambda = \frac{h}{\gamma ~m~v}$
$\gamma~v = \frac{h}{m~\lambda}$
$\frac{1}{\sqrt{1-v^2/c^2}}~v = \frac{h}{m~\lambda}$
$\frac{v^2}{1-v^2/c^2} = (\frac{h}{m~\lambda})^2$
$v^2~c^2 = (\frac{h}{m~\lambda})^2(c^2-v^2)$
$v^2~[c^2+(\frac{h}{m~\lambda})^2] = (\frac{h~c}{m~\lambda})^2$
$v^2 = \frac{(\frac{h~c}{m~\lambda})^2}{c^2+(\frac{h}{m~\lambda})^2}$
$v^2 = \frac{(h~c)^2}{(m~\lambda~c)^2+h^2}$
$v = \frac{h~c}{\sqrt{(m~\lambda~c)^2+h^2}}$
$v = \frac{(6.626\times 10^{-34}~J~s)(3.0\times 10^8~m/s)}{\sqrt{[(9.109\times 10^{-31}~kg)~(1.0\times 10^{-12}~m)~(3.0\times 10^8~m/s)]^2+(6.626\times 10^{-34}~J~s)^2}}$
$v = 2.8\times 10^8~m/s$
(b) We can find the speed:
$\lambda = \frac{h}{m~v}$
$v = \frac{h}{m~\lambda}$
$v = \frac{6.626\times 10^{-34}~J~s}{(9.109\times 10^{-31}~kg)~(1.0\times 10^{-9}~m)}$
$v = 7.3\times 10^5~m/s$
(c) We can find the speed:
$\lambda = \frac{h}{m~v}$
$v = \frac{h}{m~\lambda}$
$v = \frac{6.626\times 10^{-34}~J~s}{(9.109\times 10^{-31}~kg)~(1.0\times 10^{-6}~m)}$
$v = 730~m/s$
(d) We can find the speed:
$\lambda = \frac{h}{m~v}$
$v = \frac{h}{m~\lambda}$
$v = \frac{6.626\times 10^{-34}~J~s}{(9.109\times 10^{-31}~kg)~(1.0\times 10^{-3}~m)}$
$v = 0.73~m/s$