# How do I find the distance travelled in time $t$ in following question? [closed]

The particle starts from the origin with an initial velocity $u$ and the acceleration of the particle is increasing linearly with time $t$ as $bt$. Now what will the distance traveled by particle in the time $t$ be?

## closed as off-topic by Brian Moths, CuriousOne, ACuriousMind♦, Qmechanic♦Feb 16 '16 at 8:11

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Acceleration is the second derivative of position, so if the acceleration is equal to $bt$ then:

$$\frac{d^2x}{dt^2} = bt$$

You simply need to solve this differential equation and use the initial conditions you're provided. In this case you can use a technique called ansatz (which basically means guessing). Suppose you have some equation:

$$x = At^3 + Bt^2 + Ct + D$$

then:

$$\frac{dx}{dt} = 3At^2 + 2Bt + C$$

and:

$$\frac{d^2x}{dt^2} = 6At + 2B$$

Could you find values for $A$, $B$, $C$ and $D$ that would make this solve your problem?

Homework type question. Quick answer is - As the acceleration increases linearly, you can consider a uniform acceleration of $a = \frac{bt}{2}$, and then use $s = ut + \frac{1}{2}at^2$