A classical mechanics problem by Nitin Sachan

Classical Mechanics Level 4

A particle of mass m m moves on the x x -axis as follows:

  • It starts from rest at t = 0 t=0 from the point x = 0 x=0 and comes to rest at t = 1 t=1 at the point x = 1 x=1 .
  • No other information is available about its motion at intermediate times ( 0 < t < 1 ) (0<t<1) .
  • If α \alpha denotes the instantaneous acceleration of the particle.
  • It is given that at some point or points in the path ∣ α ∣ ≥ K |\alpha | \geq K , where K K is a natural number.

Compute K K .


The answer is 4.

Nitin Sachan by Nitin Sachan , 37, India

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1 solution

Andre Castagna
Oct 28, 2017

The acceleration profile must be symmetric about the time axis since the net change in velocity is zero. The profile that given the smallest maximum acceleration will be a step function starting at height K at t=0 and changing to -K at t = 1 2 \frac{1}{2} . Working with just the interval t = [0, 1 2 \frac{1}{2} ],

v(t) = ∫ \int (K dt), with v(0) = 0.

v(t) = K t

s( 1 2 \frac{1}{2} ) = ∫ \int (K t) from 0 to 1 2 \frac{1}{2}

= K 8 \frac{K}{8} .

Since s( 1 2 \frac{1}{2} ) = 1 2 \frac{1}{2} , we have K 8 \frac{K}{8} = 1 2 \frac{1}{2} .

K = 4

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How did you get when the maximum acceleration will be possible

Navin Murarka - 3 years, 7 months ago

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