Question

A bus appears to go with a speed of 25 km/ hr to a car driver driving at a rate of 7km/hr northwards. if the bus actually travels in east direction then what will be its speed??

Answer 1

Let
$\vec{v}_{bus}=v\hat{i}\\\vec{v}_{car}=7\hat{j}km/h\\\text{Given},|\vec{v}_{bus}-\vec{v}_{car}|=25km/h\\\Rightarrow |v\hat{i}-7\hat{j}|^2=25^2\\\Rightarrow v^2+7^2=25^2\\\Rightarrow v=\sqrt{25^2-7^2}=24km/h$

Answer 2

Velocity of bus with respect to car is 25km/hr

Velocity of car = 7 km/hr

Now since the speed of bus is with respect to car and is in vectors there actual velocity shall be different

So, $V_b + V_c=25$

Taking square on both sides

${ \Rightarrow V_{ b } }^{ 2 }+{ 7 }^{ 2 }={ 25 }^{ 2 }$

$\Rightarrow { V_{ b } }^{ 2 }={ 25 }^{ 2 }-{ 7 }^{ 2 } { \Rightarrow V_{ b } }^{ 2 }=625-49$

$\Rightarrow V_{ b }=\sqrt { 625-49 } \Rightarrow V_{ b }=\sqrt { 576 }$  $\Rightarrow V_b = 24 km/hr$

Hence the actual velocity of car is 7km/hr and actual velocity of bus is 24 km/hr