Question About Kinematics

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2010-12-05 4:11 am

A paritcle of mass m is moving along a straight line OX under the action of a constant force 4m towards O and a retarding force of 3mv when the speed is v. If when t = 0, x = 0 and v = 3, find the displacement from O when the particle first comes to rest.

ANS : 1 - 4/9 ln(13/4)

回答 (1)

天同

2010-12-05 9:44 am

✔ 最佳答案

Acceleration a = net force/mass = -(4m + 3mv)/m = -(4 + 3v )
hence, dv/dt = -(4 +3v)
integral[dv/(4+3v)] = integral[-dt]
ln(4+3v) = -3t + C, where C is the integration constant.

Using the initial condition when t = 0 s, v = 3 m/s
we get, C = ln(13)
thus, ln(4+3v) = ln(13) - 3t ---------- (1)
when the particle comes to rest, v = 0 m/s
ln(4) = ln(13) - 3t
i.e. t = (1/3).ln(13/4)

from (1): lm[(4+3v)/13] = -3t
(4+3v) = 13.exp(-3t)
v = (1/3).[13exp(-3t) - 4]
hence, ds/dt = (1/3).[13exp(-3t) - 4]
integrate on both sides with respect to t
we get, s = (1/3).[(-13/3).exp(-3t) -4t + 13/3] ------------------ (2)

since the particle comes to rest at t = (1/3).ln(13/4), substitute the value of t into (2) and simplifying, the displacement s is found to be 1 - (4/9).ln(13/4)

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