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MCQEasyJEE 2023Velocity & Acceleration

JEE Physics 2023 Question with Solution

A car travels a distance of 'xx' with speed V1V_1 and then the same distance 'xx' with speed V2V_2 in the same direction. The average speed of the car is:

  • A

    V1V22(V1+V2)\frac{V_1 V_2}{2(V_1 + V_2)}

  • B

    V1+V22\frac{V_1 + V_2}{2}

  • C

    2xV1+V2\frac{2x}{V_1 + V_2}

  • D

    2V1V2V1+V2\frac{2 V_1 V_2}{V_1 + V_2}

Answer

Correct answer:D

Step-by-step solution

Standard Method

Given: The car covers equal distances xx and xx with speeds V1V_1 and V2V_2 respectively.

Find: The average speed of the car.

Average speed is total distance divided by total time.

d=x+x=2xd = x + x = 2xt=xV1+xV2t = \frac{x}{V_1} + \frac{x}{V_2}vavg=Total distanceTotal time=2xxV1+xV2v_{\text{avg}} = \frac{\text{Total distance}}{\text{Total time}} = \frac{2x}{\frac{x}{V_1} + \frac{x}{V_2}}vavg=2xx(1V1+1V2)=21V1+1V2v_{\text{avg}} = \frac{2x}{x\left(\frac{1}{V_1} + \frac{1}{V_2}\right)} = \frac{2}{\frac{1}{V_1} + \frac{1}{V_2}}vavg=2V1V2V1+V2v_{\text{avg}} = \frac{2V_1V_2}{V_1 + V_2}

Therefore, the average speed is 2V1V2V1+V2\frac{2V_1V_2}{V_1 + V_2}. The correct option is D. The solution labels the option as C, but the derived expression matches option D in the given options.

Equal Distance Shortcut

Given: Two equal distances are covered with speeds V1V_1 and V2V_2.

Find: The average speed.

For equal distances, average speed is the harmonic mean of the two speeds:

vavg=2V1V2V1+V2v_{\text{avg}} = \frac{2V_1V_2}{V_1 + V_2}

This works because time depends inversely on speed, so the arithmetic mean is not applicable here.

Therefore, the correct option is D.

Common mistakes

  • Using the arithmetic mean V1+V22\frac{V_1 + V_2}{2} as the average speed. This is wrong because the car covers equal distances, not equal times. Use total distance divided by total time, which gives the harmonic mean.

  • Choosing 2xV1+V2\frac{2x}{V_1 + V_2} as the average speed. This is wrong because this expression has the dimensions of time, not speed. Always check units after simplification.

  • Adding speeds directly before computing time. This is wrong because time for each segment must be found separately as distance divided by speed. First calculate xV1\frac{x}{V_1} and xV2\frac{x}{V_2}, then add them.

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