Question

A constant force of 5 N is acting on a body. If x(t) is the dispacement at time t, v(t) velocity at time t, m mass of the body, which of the following differential equations represent the situation?

• A. $m\frac{\mathrm{dx}\left(t\right)}{\mathrm{dt}}=5$
• B. $m\frac{\mathrm{dv}\left(t\right)}{\mathrm{dt}}=5$
• C. $m\frac{d^{2}x\left(t\right)}{{\mathrm{dt}}^2}=5$
• D. $5\frac{\mathrm{dx}\left(t\right)}{\mathrm{dt}}=m$

Answer 1

Answer: (B), (C)

$m\frac{d^{2}x\left(t\right)}{{\mathrm{dt}}^2}=5$

Using the second law of motion F=ma, and acceleration a is the rate of change of velocity i.e. $a=\frac{\mathrm{dv}\left(t\right)}{\mathrm{dt}}$ and velocity is the rate of change of displacement i.e.
$v=\frac{\mathrm{dx}\left(t\right)}{\mathrm{dt}}$.
So $a=\frac{\mathrm{dv}}{\mathrm{dt}}=\frac{d}{\mathrm{dt}}\left(\frac{\mathrm{dx}}{\mathrm{dt}}\right)=\frac{d^{2}x}{{\mathrm{dt}}^2}$