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Two particle are moving along two long straight lines, in the same plane with samespeed equal to 120cms. The angle between the two lines is 060 and their intersection point is O . At a certain moment, the two particles are located at distances 3m and 4m from O and are moving towardsO . Subsequently the shortest distance between them will be1) 50cm 2) 40 2cm 3) 50 2cm 4) 50 3cm |
| Answer» <html><body><p><strong>Answer:</strong></p><p>time t=0,the first particle is 300cm from O and moving toward O at a velocity of 20cm/s. </p><p>The distance of this particle to O as a function of time is: </p><p>A=300−20t</p><p>At the time t=0, the second particle is <a href="https://interviewquestions.tuteehub.com/tag/400cm-315718" style="font-weight:bold;" target="_blank" title="Click to know more about 400CM">400CM</a> from O and moving toward O at a velocity of 20m/s.</p><p></p><p></p><p>The distance of this particle to O as a function of time is: </p><p>B=400−20t</p><p>The distance between the 2 particles can be found using the cosine rule </p><p>C </p><p>2</p><p> =A </p><p>2</p><p> +B </p><p>2</p><p> −2ABcos60 </p><p>0</p><p> </p><p>C </p><p>2</p><p> =(300−20t) </p><p>2</p><p> +(400−20t) </p><p>2</p><p> −2(300−20t)(400−20t)cos60 </p><p>0</p><p> </p><p>C </p><p>2</p><p> =(400t </p><p>2</p><p> −12000t+90000)+(400t </p><p>2</p><p> −16000t+160000)−(400t </p><p>2</p><p> −14000t+120000)</p><p>C </p><p>2</p><p> =400t </p><p>2</p><p> −14000t+130000 </p><p>C= </p><p>(400t </p><p>2</p><p> −14000t+130000)</p><p> </p><p> </p><p>Now one needs to find t such that C will be minimized.</p><p>If C is minimized at a certain t, then also C </p><p>2</p><p> will be minimized at that t. </p><p>So <a href="https://interviewquestions.tuteehub.com/tag/instead-516622" style="font-weight:bold;" target="_blank" title="Click to know more about INSTEAD">INSTEAD</a> of finding the first derivative of C (which is a square root,</p><p>so a lot of work), one can simply find the first derivative of C </p><p>2</p><p> (a </p><p>quadratic expression, which is a lot <a href="https://interviewquestions.tuteehub.com/tag/less-1071906" style="font-weight:bold;" target="_blank" title="Click to know more about LESS">LESS</a> work) </p><p></p><p>C </p><p>2</p><p> =400t </p><p>2</p><p> −14000t+130000 </p><p>dt </p><p>2</p><p> </p><p>dC </p><p>2</p><p> </p><p> </p><p> =800t−14000 </p><p>800t−14000=0 </p><p>t=17.5s </p><p>At time t=17.5s the distance between the 2 particles is minimized. </p><p>The distance at that time will be: </p><p>C= </p><p>(400×17.5 </p><p>2</p><p> −14000×17.5+130000)</p><p> </p><p> </p><p>C= </p><p><a href="https://interviewquestions.tuteehub.com/tag/7500-335415" style="font-weight:bold;" target="_blank" title="Click to know more about 7500">7500</a></p><p> </p><p> </p><p>⇒C=50 </p><p>3</p><p> </p><p> </p><p></p><p></p></body></html> | |