Multiple choice

The two ends of a train moving with uniform acceleration pass a certain point with velocities u and v. The velocity with which the middle point of the train passes the same point is?

  1. $\dfrac{v+u}{2}$
  2. $\dfrac{u^2+v^2}{2}$
  3. $\sqrt{\dfrac{u^2+v^2}{2}}$
  4. $\sqrt{v+u}$
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C Correct answer
Explanation

Using the kinematic equation v^2 = u^2 + 2as, the velocity v_m at the midpoint s/2 is v_m^2 = u^2 + 2a(s/2) = u^2 + as. Since v^2 = u^2 + 2as, we have as = (v^2 - u^2)/2. Substituting this, v_m^2 = u^2 + (v^2 - u^2)/2 = (u^2 + v^2)/2. Thus, v_m = sqrt((u^2 + v^2)/2).

AI explanation

Let the length of the train be L and its uniform acceleration be a. When the front end of the train passes the point, its velocity is u, so u^2 = 2as. When the rear end passes the point, the train has moved a distance L, making the velocity v, so v^2 = u^2 + 2aL. When the middle of the train passes the point, the train has moved a distance of L/2, giving a velocity w where w^2 = u^2 + 2a(L/2) = u^2 + aL. Substituting aL from the previous equations gives w^2 = u^2 + (v^2 - u^2) / 2, which simplifies to w^2 = (u^2 + v^2) / 2. Taking the square root of both sides results in the velocity w = sqrt((u^2 + v^2) / 2).

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