# 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

This question appears to be off-topic. The users who voted to close gave this specific reason:

• "Homework-like questions should ask about a specific physics concept and show some effort to work through the problem. We want our questions to be useful to the broader community, and to future users. See our meta site for more guidance on how to edit your question to make it better" – Brian Moths, CuriousOne, ACuriousMind, Qmechanic
If this question can be reworded to fit the rules in the help center, please edit the question.

## 2 Answers

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 $s = ut + \frac{1}{2}at^2$