Answer
427.8k+ views
Hint: The given case is in a two dimensional system, therefore the directional vectors should also be considered. The total magnitude of the acceleration depends on the acceleration among different directions.
Complete step by step answer:
When the two cars are moving in the different directions then the relative velocity of the cars is calculated by subtracting the velocities of the car. When the car moves in directly opposite directions then the relative velocity of the car is calculated by the sum of two velocities.
The direction of the cars is as shown:
The formula to calculate the relative acceleration of car B with respect to car A is
${a_{BA}} = {a_B}\hat j - {a_A}\hat i$
Here, ${a_{BA}}$ is the relative acceleration of car A with respect to car B, ${a_A}$ is the acceleration of car A, $\hat i$ is the unit vector along east direction, ${a_B}$is the acceleration of car B and $\hat j$ is the unit vector along north direction.
Substitute $2\,{{\rm{m}} {\left/{\vphantom {{\rm{m}} {{{\rm{s}}^{\rm{2}}}}}} \right.
} {{{\rm{s}}^{\rm{2}}}}}$ for ${a_A}$ and $4\,{{\rm{m}} {\left/{\vphantom {{\rm{m}} {{{\rm{s}}^{\rm{2}}}}}} \right.} {{{\rm{s}}^{\rm{2}}}}}$for ${a_B}$ in the formula to calculate the relative acceleration of the car B with respect to car A.
${a_{BA}} = \left( {2\,{{\rm{m}} {\left/ {\vphantom {{\rm{m}} {{{\rm{s}}^{\rm{2}}}}}} \right.} {{{\rm{s}}^{\rm{2}}}}}} \right)\hat j - \left( {4\,{{\rm{m}} {\left/{\vphantom {{\rm{m}} {{{\rm{s}}^{\rm{2}}}}}} \right.} {{{\rm{s}}^{\rm{2}}}}}} \right)\hat i$
The formula to calculate the magnitude of relative acceleration of car B with respect to car A is
$\left| {{a_{BA}}} \right| = \sqrt {a_B^2 + a_A^2} \,\,$
Substitute $2\,{{\rm{m}} {\left/{\vphantom {{\rm{m}} {{{\rm{s}}^{\rm{2}}}}}} \right. } {{{\rm{s}}^{\rm{2}}}}}$ for ${a_A}$ and $4\,{{\rm{m}} {\left/{\vphantom {{\rm{m}} {{{\rm{s}}^{\rm{2}}}}}} \right.} {{{\rm{s}}^{\rm{2}}}}}$for ${a_B}$ in the formula to calculate magnitude of the relative acceleration of the car B with respect to car A.
$ \left| {{a_{BA}}} \right| = \sqrt {{{\left( {2\,{{\rm{m}} {\left/{\vphantom {{\rm{m}} {{{\rm{s}}^{\rm{2}}}}}} \right. } {{{\rm{s}}^{\rm{2}}}}}} \right)}^2} + {{\left( {4\,{{\rm{m}} {\left/ {\vphantom {{\rm{m}} {{{\rm{s}}^{\rm{2}}}}}} \right.} {{{\rm{s}}^{\rm{2}}}}}} \right)}^2}} \\
= \sqrt {4 + 16} \,{{\rm{m}} {\left/ {\vphantom {{\rm{m}} {{{\rm{s}}^{\rm{2}}}}}} \right. } {{{\rm{s}}^{\rm{2}}}}}\\
= \sqrt {20} \,{{\rm{m}} {\left/ {\vphantom {{\rm{m}} {{{\rm{s}}^{\rm{2}}}}}} \right.} {{{\rm{s}}^{\rm{2}}}}}\\
= 4.47\,{{\rm{m}} {\left/
{\vphantom {{\rm{m}} {{{\rm{s}}^{\rm{2}}}}}} \right.} {{{\rm{s}}^{\rm{2}}}}} $
Thus, the relative acceleration of car B with respect to car A is $4.47\,{{\rm{m}} {\left/ {\vphantom {{\rm{m}} {{{\rm{s}}^{\rm{2}}}}}} \right.} {{{\rm{s}}^{\rm{2}}}}}$.
Note:
The direction of car B relative to car A can also be found from this example because the direction of the car can be calculated using the acceleration of the cars in different directions.
Complete step by step answer:
When the two cars are moving in the different directions then the relative velocity of the cars is calculated by subtracting the velocities of the car. When the car moves in directly opposite directions then the relative velocity of the car is calculated by the sum of two velocities.
The direction of the cars is as shown:
![seo images](https://www.vedantu.com/question-sets/9c3911cb-86e3-4d06-a6e1-ddb39474b97d5390904349234404376.png)
The formula to calculate the relative acceleration of car B with respect to car A is
${a_{BA}} = {a_B}\hat j - {a_A}\hat i$
Here, ${a_{BA}}$ is the relative acceleration of car A with respect to car B, ${a_A}$ is the acceleration of car A, $\hat i$ is the unit vector along east direction, ${a_B}$is the acceleration of car B and $\hat j$ is the unit vector along north direction.
Substitute $2\,{{\rm{m}} {\left/{\vphantom {{\rm{m}} {{{\rm{s}}^{\rm{2}}}}}} \right.
} {{{\rm{s}}^{\rm{2}}}}}$ for ${a_A}$ and $4\,{{\rm{m}} {\left/{\vphantom {{\rm{m}} {{{\rm{s}}^{\rm{2}}}}}} \right.} {{{\rm{s}}^{\rm{2}}}}}$for ${a_B}$ in the formula to calculate the relative acceleration of the car B with respect to car A.
${a_{BA}} = \left( {2\,{{\rm{m}} {\left/ {\vphantom {{\rm{m}} {{{\rm{s}}^{\rm{2}}}}}} \right.} {{{\rm{s}}^{\rm{2}}}}}} \right)\hat j - \left( {4\,{{\rm{m}} {\left/{\vphantom {{\rm{m}} {{{\rm{s}}^{\rm{2}}}}}} \right.} {{{\rm{s}}^{\rm{2}}}}}} \right)\hat i$
The formula to calculate the magnitude of relative acceleration of car B with respect to car A is
$\left| {{a_{BA}}} \right| = \sqrt {a_B^2 + a_A^2} \,\,$
Substitute $2\,{{\rm{m}} {\left/{\vphantom {{\rm{m}} {{{\rm{s}}^{\rm{2}}}}}} \right. } {{{\rm{s}}^{\rm{2}}}}}$ for ${a_A}$ and $4\,{{\rm{m}} {\left/{\vphantom {{\rm{m}} {{{\rm{s}}^{\rm{2}}}}}} \right.} {{{\rm{s}}^{\rm{2}}}}}$for ${a_B}$ in the formula to calculate magnitude of the relative acceleration of the car B with respect to car A.
$ \left| {{a_{BA}}} \right| = \sqrt {{{\left( {2\,{{\rm{m}} {\left/{\vphantom {{\rm{m}} {{{\rm{s}}^{\rm{2}}}}}} \right. } {{{\rm{s}}^{\rm{2}}}}}} \right)}^2} + {{\left( {4\,{{\rm{m}} {\left/ {\vphantom {{\rm{m}} {{{\rm{s}}^{\rm{2}}}}}} \right.} {{{\rm{s}}^{\rm{2}}}}}} \right)}^2}} \\
= \sqrt {4 + 16} \,{{\rm{m}} {\left/ {\vphantom {{\rm{m}} {{{\rm{s}}^{\rm{2}}}}}} \right. } {{{\rm{s}}^{\rm{2}}}}}\\
= \sqrt {20} \,{{\rm{m}} {\left/ {\vphantom {{\rm{m}} {{{\rm{s}}^{\rm{2}}}}}} \right.} {{{\rm{s}}^{\rm{2}}}}}\\
= 4.47\,{{\rm{m}} {\left/
{\vphantom {{\rm{m}} {{{\rm{s}}^{\rm{2}}}}}} \right.} {{{\rm{s}}^{\rm{2}}}}} $
Thus, the relative acceleration of car B with respect to car A is $4.47\,{{\rm{m}} {\left/ {\vphantom {{\rm{m}} {{{\rm{s}}^{\rm{2}}}}}} \right.} {{{\rm{s}}^{\rm{2}}}}}$.
Note:
The direction of car B relative to car A can also be found from this example because the direction of the car can be calculated using the acceleration of the cars in different directions.
Recently Updated Pages
Mark and label the given geoinformation on the outline class 11 social science CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
When people say No pun intended what does that mea class 8 english CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Name the states which share their boundary with Indias class 9 social science CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Give an account of the Northern Plains of India class 9 social science CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Change the following sentences into negative and interrogative class 10 english CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Advantages and disadvantages of science
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Trending doubts
Bimbisara was the founder of dynasty A Nanda B Haryanka class 6 social science CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Which are the Top 10 Largest Countries of the World?
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Differentiate between homogeneous and heterogeneous class 12 chemistry CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
10 examples of evaporation in daily life with explanations
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Fill the blanks with the suitable prepositions 1 The class 9 english CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Give 10 examples for herbs , shrubs , climbers , creepers
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
How do you graph the function fx 4x class 9 maths CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Difference Between Plant Cell and Animal Cell
![arrow-right](/cdn/images/seo-templates/arrow-right.png)