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Motion in A Straight Line question

2023 · 25 Jan · Shift 1 · Q55
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  5. /2023 · 25 Jan · Shift 1 · Q55

Motion in A Straight Line question

2023 · 25 Jan · Shift 1 · Q55

JEE MainPhysicsMotion in A Straight LineMCQ+4 / −1
A car travels a distance of 'xxx' with speed v1v_1v1​ and then same distance 'xxx' with speed v2v_2v2​ in the same direction. The average speed of the car is :
  1. A
    v1v22(v1+v2){{{v_1}{v_2}} \over {2({v_1} + {v_2})}}2(v1​+v2​)v1​v2​​
  2. B
    2v1v2v1+v2{{2{v_1}{v_2}} \over {{v_1} + {v_2}}}v1​+v2​2v1​v2​​
  3. C
    2xv1+v2{{2x} \over {{v_1} + {v_2}}}v1​+v2​2x​
  4. D
    v1+v22{{{v_1} + {v_2}} \over 2}2v1​+v2​​
View written solutionFree

Correct answer: B

  1. Given:

    • First distance =x= x=x traveled with speed v1v_1v1​
    • Second distance =x= x=x traveled with speed v2v_2v2​

  2. Total distance traveled: x+x=2xx + x = 2xx+x=2x

  3. Time taken for each part:

    • For first part, t1=xv1t_1 = \frac{x}{v_1}t1​=v1​x​
    • For second part, t2=xv2t_2 = \frac{x}{v_2}t2​=v2​x​

  4. Total time: t=t1+t2=xv1+xv2t = t_1 + t_2 = \frac{x}{v_1} + \frac{x}{v_2}t=t1​+t2​=v1​x​+v2​x​ t=x(1v1+1v2)t = x\left(\frac{1}{v_1} + \frac{1}{v_2}\right)t=x(v1​1​+v2​1​) t=x(v1+v2v1v2)t = x\left(\frac{v_1+v_2}{v_1v_2}\right)t=x(v1​v2​v1​+v2​​)

  5. Average speed formula: vavg=total distancetotal timev_{avg} = \frac{\text{total distance}}{\text{total time}}vavg​=total timetotal distance​ vavg=2xx(v1+v2v1v2)v_{avg} = \frac{2x}{x\left(\frac{v_1+v_2}{v_1v_2}\right)}vavg​=x(v1​v2​v1​+v2​​)2x​

  6. Cancel xxx: vavg=2(v1+v2v1v2)v_{avg} = \frac{2}{\left(\frac{v_1+v_2}{v_1v_2}\right)}vavg​=(v1​v2​v1​+v2​​)2​ vavg=2v1v2v1+v2v_{avg} = \frac{2v_1v_2}{v_1+v_2}vavg​=v1​+v2​2v1​v2​​

  7. Match with options: 2v1v2v1+v2\boxed{\frac{2v_1v_2}{v_1+v_2}}v1​+v2​2v1​v2​​​ This corresponds to Option B.

  8. Option check:

    • A: Incorrect
    • B: Correct
    • C: Incorrect (has dimensions of time, not speed)
    • D: Incorrect for equal distances; this is arithmetic mean, not average speed here

Therefore, the average speed is 2v1v2v1+v2\boxed{\frac{2v_1v_2}{v_1+v_2}}v1​+v2​2v1​v2​​​

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