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This is the second of our many quizzes on Relative velocity. Click on this to access more such competitive quizzes on Mechanics.

Relative velocity Quiz 2

physics

 A person is travelling towards the positive x-direction at a speed of $3$ m/s and a bicycle is moving towards the positive $y$ direction at a speed of $15$ m/s. Calculate the relative velocity of the man with respect to the bicycle. 

The relative velocity of the man with respect to the bicycle is given by ${{v}_{pb}}=\sqrt{{{v}_{p}}^{2}-{{v}_{b}}^{2}}$ and angle can be calculated by taking the tan of the angle $\tan \theta =\dfrac{{{v}_{b}}}{{{v}_{p}}}$. 

  <img  src="https://cmds-vedantu.s3.ap-southeast-1.amazonaws.com/prod/question-sets/c3dcc22e-21eb-47d5-9be5-6cb1cab938357758132610133643897.png"/><br>$ \\ \text{Consider the velocity of the person be}\, {{v}_{p}} \\ \text{and velocity of the bicycle be}\,{{v}_{b}}\,\text{as shown in the figure.} \\ \text{So, the relative velocity of the man with respect to the bicycle will be,}\\ {{v}_{pb}}=\sqrt{{{v}_{p}}^{2}+{{v}_{b}}^{2}}….(i) \\ \text{Here}\,{{v}_{p}}=3\,m/s \text{and}\,{{v}_{b}}=15\,m/s\,\text{so putting this in equation (i), we get,}\\ {{v}_{pb}}=\sqrt{{{3}^{2}}+{{15}^{2}}} \\ \Rightarrow{{v}_{pb}}=15.3\,m/s \\ \text{For calculating the angle we take the tan of the angle,}\\ \tan \theta =\dfrac{{{v}_{b}}}{{{v}_{p}}}=\dfrac{15}{3} \\ \therefore \theta ={{\tan }^{-1}}(\dfrac{15}{3})={{78.7}^{0}} \\ \text{Therefore, the relative velocity of the man with respect to the bicycle is 15.3 m/s}\\ \text{and}\,{{78.7}^{0}}\,\text{northeast of the direction of motion of the person.}$

The velocity of a fly in reference frame $A$ is $12\hat{i}+6\hat{j}$ and the relative velocity of reference frame $A$ with respect to reference frame $B$ is $3\hat{j}$. What is the velocity of the fly in reference frame $B$? 

Relative velocity of object $A$ with respect to another object $B$ is given by ${{v}_{ab}}={{v}_{a}}-{{v}_{b}}$.Put in the values given in the question to find the relative velocity. 

  $\text{ Consider the fly in frame A as moving object F,}\\ \text{its relative velocity with respect to reference frame B is given by,}\\ {{v}_{FB}}={{v}_{F}}-{{v}_{B}}….(i)\\ \text{According to the question,}\\ {{v}_{F}}=12\hat{i}+6\hat{j}\,\text{and}\,{{v}_{B}}=3\hat{j}\\ \text{Putting the above values in equation (i), we get,}\\ {{v}_{FB}}=12\hat{i}+6\hat{j}-3\hat{j}\\ \therefore{{v}_{FB}}=12\hat{i}+3\hat{j}$

 A person is sitting on a sleigh pulled by dogs moving at constant velocity $6$ m/s. If the person throws a ball forward at a velocity of $1.2$ m/s then calculate the relative velocity of the ball with respect to a stationary observer? 

Relative velocity of object $A$ with respect to another object $B$ is given by ${{v}_{ab}}={{v}_{a}}-{{v}_{b}}$.Put in the values given in the question to find the relative velocity. If motion of $A$ and $B$ is opposite to each other then we add the velocities to get the relative velocity. 

 $\text{Consider the person on the sleigh as object A,}\\ \text{its relative velocity with respect to the ball thrown as object B is given by,}\\ {{v}_{ab}}={{v}_{a}}-{{v}_{b}}….(i)\\ \text{According to the question,}\\ {{v}_{a}}=6\,m/s\,\text{and}\,{{v}_{b}}=-1.2\,m/s\,(\text{Minus sign because their motion is in the opposite to each other})\\ \text{Putting the above values in equation (i), we get,}\\ {{v}_{ab}}=6-(-1.2)\\ \therefore{{v}_{ab}}=7.2\,m/s$

A space shuttle is moving towards earth at a constant velocity of $0.3\,c$, it sends a light signal to earth at the velocity of $3\times {{10}^{8}}$ m/s. What is the relative velocity of the laser with respect to earth? $(\text{Here, c is the speed of light})$ 

The relative velocity of the laser with respect to earth is $v=\dfrac{{{v}_{s}}+{{v}_{l}}}{1+\dfrac{{{v}_{s}}{{v}_{l}}}{{{c}^{2}}}}$. Put the values given in the question and solve to find the relative velocity. 

 $\text{Since, laser light is moving relative to speed of light,}\\ \text{the relative velocity of the laser with respect to earth is,}\\ v=\dfrac{{{v}_{s}}+{{v}_{l}}}{1+\dfrac{{{v}_{s}}{{v}_{l}}}{{{c}^{2}}}}….(i)\\ \text{According to the question,}\\ {{v}_{s}}=0.3c\,\text{and}\,{{v}_{l}}=3\times {{10}^{8}}\,m/s=c\\ \text{Putting the above values in equation (i), we get,}\\ v=\dfrac{0.3c+c}{1+\dfrac{0.3c\times c}{{{c}^{2}}}}\\ \Rightarrow v=\dfrac{1.3c}{1.3}\\ \therefore v=c$

If a honey bee has a velocity of $7$ m/s when no wind is blowing and it takes $5$ seconds to travel $50$ m straight with a wind against its motion, then what is the velocity of the wind? 

The relative velocity of the honey bee with respect to wind is given by ${{v}_{bw}}={{v}_{b}}-{{v}_{w}}$.Put in the known values and find the velocity of the wind. 

 $\text{The relative velocity of the honey bee with respect to wind is given by,}\\ {{v}_{bw}}={{v}_{b}}-{{v}_{w}}….(i)\\ \text{According to the question, if it takes 5 seconds for the bee}\\ \text{to travel 50 m straight with respect to the wind, then,}\\ {{v}_{bw}}=\dfrac{D}{T}\\ \Rightarrow{{v}_{bw}} =\dfrac{50}{5} \\ \Rightarrow{{v}_{bw}} =10\,m/s \\ \text{And}\,{{v}_{b}}=7\,m/s\\ \text{Putting the above values in equation (i), we get, the speed of the wind as,}\\ {{v}_{w}}=7-10\\ \therefore{{v}_{w}}=-3\,m/s…(\text{Negative sign because wind is against the direction of motion of the bee})$