Question

Two particles A and B start moving with velocities 20m/s and 30√2m/s along x-axis and at an angle of 45 degree with x-axis respectively in xy - plane from origin. what is the relative velocity of B w.r.t A ?

Answer 1

Answer:

The relative velocity is 31.62 m/s

Explanation:

Given that,

Velocity of particle A $v_{A} = 20\ m/s$

Velocity of particle B $v_{B} = 30\sqrt{2}\ m/s$

Angle $\theta=45^{0}$

The component of particle A along x-axis

$V_{A}_{x}=20 m/s$

The component of particle A along y-axis

$V_{A}_{y}=0 m/s$

The component of particle B along x-axis

$V_{B}_{x}=30\sqrt{2}cos 45^{0} m/s$

$V_{B}_{x}=30\ m/s$

The component of particle B along y-axis

$V_{B}_{y}=30\sqrt{2}sin45^{0} m/s$

$V_{B}_{y}=30\ m/s$

The velocity of AB along x-axis

$v_{AB}_{x}=v_{B}_{x}-v_{A}_{x}$

$v_{AB}_{x}=30-20$

$v_{AB}_{x}=10\ m/s$

The velocity of AB along y-axis

$v_{AB}_{y}=v_{B}_{y}-v_{A}_{y}$

$v_{AB}_{y}=30-0$

$v_{AB}_{y}=30\ m/s$

Now, the relative velocity of B w.r.t A

$v_{AB}_{x}= \sqrt{v_{AB}_{x}^{2}+{v_{AB}_{y}^{2}+2\times v_{AB}_{x}\times v_{AB}_{y}\times cos\alpha}$

Where, $\alph=90^{0}$

$v_{AB}_{x}= \sqrt{v_{AB}_{x}^{2}+{v_{AB}_{y}^{2}+2\times 0$

$v_{AB}_{x}= \sqrt{10^{2}+30^{2}}$

$v_{AB}_{x}= 31.62\ m/s$

Hence, The relative velocity is 31.62 m/s

Answer 2

According to the given data, Velocity of B in x – direction at angle of 45 gives the component as

$V_{ b }=30\sqrt { 2 } \times cos45$

$\Rightarrow v_{ b }=30\frac { m }{ s }$

Same for along y axis be

$v_{ b }=30\sqrt { 2 } \times sin45$

$\Rightarrow v_{ b }=30\frac { m }{ s }$

Velocity of A which is along x axis $v_{ a }=20\frac { m }{ s }$ and is zero for y component.

So the $V _{ab}$ along x axis  $= v_b - v _a$ along x – axis $=30-20=10\frac { m }{ s }$

And for y axis be $v_{ ab }=v_{ b }-v_{ a }=30-0=30\frac { m }{ s }$

Thereby the relative velocity be

$V_{ ab }=\sqrt { { { V }_{ { ab }_{ x } } }^{ 2 }+{ { V }_{ { ab }_{ y } } }^{ 2 }+2\times { V }_{ { ab }_{ x } }\times { V }_{ { ab }_{ y } }cos\alpha }$

Where $\alpha = 90 \quad degree$

$\Rightarrow V_{ ab }=\sqrt { { { V }_{ { ab }_{ x } } }^{ 2 }+{ { V }_{ { ab }_{ y } } }^{ 2 }+0 }$

$\Rightarrow V_{ ab }=\sqrt { { 10 }^{ 2 }+{ 30 }^{ 2 } } =\sqrt { 1000 }$

$\Rightarrow V_{ ab }=31.625\frac { m }{ s }$