Kinematics

Classical Mechanics Level 1

The initial acceleration of a particle moving in a straight line is a 0 a_0 and its initial velocity is zero. The acceleration reduces continuously to half in every t 0 t_0 seconds. The terminal speed of the particle is __________ . \text{\_\_\_\_\_\_\_\_\_\_}.

a 0 t 0 2 \frac{a_0 t_0}2 a 0 t 0 a_0 t_0 a 0 t 0 ln ⁡ ( 2 ) \frac{a_0 t_0}{\ln (2)} a 0 t 0 ln ⁡ ( 2 ) a_0 t_0 \ln (2)

Aakhyat Singh by Aakhyat Singh , 20, India

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1 solution

Steven Chase
Oct 21, 2017

Acceleration:

a = a 0 e − λ t λ = l n ( 2 ) t 0 a = a_0 \, e^{-\lambda t} \hspace{1cm} \lambda = \frac{ln(2)}{t_0}

Integrating for velocity gives:

v = a 0 λ ( 1 − e − λ t ) v ∞ = a 0 λ = a 0 t 0 l n ( 2 ) v = \frac{a_0}{\lambda} (1 - e^{-\lambda t}) \\ v_{\infty} = \frac{a_0}{\lambda} = \frac{a_0 t_0}{ln(2)}

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But @Steven Chase sir, how do we realise that we have to use exponential functions ??

Aakhyat Singh - 3 years, 7 months ago

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This is essentially the same thing as radioactive half life. Wikipedia has a good page on it.

Steven Chase - 3 years, 7 months ago

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