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Having gone through a plank of thickness h, a bullet changed its velocity from v_0 to v. Find the time of motion of the bullet in the plank, assuming the resistance force to be proportional to the square of the velocity. |
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Answer» Solution :ACCORDING to the problem `m(DV)/(dt)=-kv^2`or, `m(dv)/(v^2)=-kdt` Integrating, within the limits, `UNDERSET(v_0)overset(v)int(dv)/(v^2)=-k/m underset(0)overset(t)intdt` or, `t=m/k((v_0-v))/(v_0v)` (1) To FIND the value of k, rewrite `mv(dv)/(ds)=-kv^2`, or, `(dv)/(v)=-k/mds` On integrating `underset(v_0)overset(v)int(dv)/(v)=-k/m underset(0)overset(h)intds` So, `k=m/h1n(v_0)/(v)` (2) Putting the value of k from (2) and (1), we get `t=(h(v_0-v))/(v_0v1nv_0/v)` |
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