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.

Having gone through a plank of thickness h, a bullet changed its veloc

1.	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.
Answer» 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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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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