A particle in an electric and magnetic field is in motion. The time equations are in polar coordinates. r=r_0 e^(−(t/b)) and θ=(t/b) and b are positive constants. 1\Calculate the vector equation of the velocity of the particle. 2\Show that the angle (v_1 ^′ ,u_0 ′) is constant, and find the value. 3\Find the vector of acceleration of the particle. 4\Show that the angle (v_1 ^→ ,u_n ′) is constant, and find it. 5\Calculate the radius of this trajectory.


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Question Number 108042 by Ar Brandon last updated on 14/Aug/20
$A particle in an electric and magnetic field is in motion. The time equations are in polar coordinates. r=r_0 e^(−(t/b)) and θ=(t/b) and b are positive constants. 1\Calculate the vector equation of the velocity of the particle. 2\Show that the angle (v_1 ^′ ,u_0 ′) is constant, and find the value. 3\Find the vector of acceleration of the particle. 4\Show that the angle (v_1 ^→ ,u_n ′) is constant, and find it. 5\Calculate the radius of this trajectory.$
$$\mathrm{A}\:\mathrm{particle}\:\mathrm{in}\:\mathrm{an}\:\mathrm{electric}\:\mathrm{and}\:\mathrm{magnetic}\:\mathrm{field}\:\mathrm{is}\:\mathrm{in}\:\mathrm{motion}. \\ $$$$\mathrm{The}\:\mathrm{time}\:\mathrm{equations}\:\mathrm{are}\:\mathrm{in}\:\mathrm{polar}\:\mathrm{coordinates}. \\ $$$$\mathrm{r}=\mathrm{r}_{\mathrm{0}} \mathrm{e}^{−\frac{\mathrm{t}}{\mathrm{b}}} \:\mathrm{and}\:\theta=\frac{\mathrm{t}}{\mathrm{b}}\:\mathrm{and}\:\mathrm{b}\:\mathrm{are}\:\mathrm{positive}\:\mathrm{constants}. \\ $$$$\mathrm{1}\backslash\mathrm{Calculate}\:\mathrm{the}\:\mathrm{vector}\:\mathrm{equation}\:\mathrm{of}\:\mathrm{the}\:\mathrm{velocity}\:\mathrm{of}\:\mathrm{the}\:\mathrm{particle}. \\ $$$$\mathrm{2}\backslash\mathrm{Show}\:\mathrm{that}\:\mathrm{the}\:\mathrm{angle}\:\left(\mathrm{v}_{\mathrm{1}} ^{'} ,\mathrm{u}_{\mathrm{0}} '\right)\:\mathrm{is}\:\mathrm{constant},\:\mathrm{and}\:\mathrm{find}\:\mathrm{the}\:\mathrm{value}. \\ $$$$\mathrm{3}\backslash\mathrm{Find}\:\mathrm{the}\:\mathrm{vector}\:\mathrm{of}\:\mathrm{acceleration}\:\mathrm{of}\:\mathrm{the}\:\mathrm{particle}. \\ $$$$\mathrm{4}\backslash\mathrm{Show}\:\mathrm{that}\:\mathrm{the}\:\mathrm{angle}\:\left(\overset{\rightarrow} {\mathrm{v}}_{\mathrm{1}} ,\mathrm{u}_{\mathrm{n}} '\right)\:\mathrm{is}\:\mathrm{constant},\:\mathrm{and}\:\mathrm{find}\:\mathrm{it}. \\ $$$$\mathrm{5}\backslash\mathrm{Calculate}\:\mathrm{the}\:\mathrm{radius}\:\mathrm{of}\:\mathrm{this}\:\mathrm{trajectory}. \\ $$
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