A particle is moving three times as fast as an electron.The ratio of the de Broglie wavelength of the particle to that of theelectron is \(1.813\times 10^{-4}\) . Calculate the particle’s mass and identify the particle.
Answer
According to the de Broglie equation, \(\lambda =\frac h p\) Planck’s constant, \(h=6.62\times 10^{-34} Js\) For a particle having mass \(m\) and velocity \(v\) . \(\lambda =\frac h{\mathit{mv}}\) \(m=\frac h{\mathit{\lambda v}}\) For an electron, \(m_e=\frac h{\lambda _ev_e}\) \(\frac{\lambda }{\lambda _e}=1.813\times 10^{-4}\) \(\frac v{v_e}=3\) Mass of the particle, \(m=m_e\times \frac{\lambda }{\lambda _e}\times \frac v{v_e}\) \(m=9.11\times 10^{-31}\times \frac 1{1.813\times 10^{-4}}\times \frac 1 3=1.675\times 10^{-27}\mathit{kg}\) This mass could be of a proton or a neutron.
A tangent is drawn at any point on the hyperbola \(\frac{x^2}{a^2}-\frac{y^2}{b^2}=1\) . If this tangent is intersected by the tangents at the vertices at points \(P\) and \(Q\) , then which of the following is/are true.( )
A. \(S,S^\prime ,P\) and \(Q\) are concyclic.
B. \(\mathit{PQ}\) is diameter of the circle.
C. \(S,S^\prime ,P\) and \(Q\) forms rhombus.
D. \(\mathit{PQ}\) is diagonal of acute angle of the rhombus formed by \(S,S^\prime ,P\) and \(Q\) .
Let \( f(x+\frac{1}{y})+f(x-\frac{1}{y})=2f\left(x\right)f\left(\frac{1}{y}\right)\forall\;x\), \( y\in\;R\)\( y\ne\;0 and\;f\left(0\right)=0\) then the value of \( f\left(1\right)+f\left(2\right)=?\) ( )
Let \( \overrightarrow{a}\),\( \overrightarrow{b}\) and \( \overrightarrow{c}\) be three vectors having magnitudes \( 1\),\( 1\) and \( 2\) respectively. If \( \overrightarrow{a}\times \left(\overrightarrow{a}\times \overrightarrow{c}\right)+\overrightarrow{b}=0,\) then the acute angle between \( \overrightarrow{a}\) and \( \overrightarrow{c}\) is ( )
Consider the function h(x) = \( h\left(x\right)=\frac{{g}^{2}\left(x\right)}{2}+3{x}^{3}-5\), where g(x) is a continuous and differentiable function. It is given that h(x) is a monotonically increasing function and g(0) = 4. Then which of the following is not true? ( )
Consider the function \( f\left(x\right)={\int }_{0}^{x}(5ln(1+{t}^{2})-10t {tan}^{-1}t+16sint)dt\). Which is not true for \( {\int }_{0}^{x}f\left(t\right)dt\)? ( )
A. Positive for all \( x\in \left(0,1\right)\)
B. Increasing for all \( x\in \left(0,1\right)\)
C. Non-monotonic for all \( x\in \left(0,1\right)\)
The gas \(X\)at \(1\) atm is bubbled through a solution containing a mixture of \(1MY^{-}\) and \(1MZ^{-}\) at \(25^{\circ }C\) . If the order of reduction potential is \(Z>Y>X\) , then ( )