Question

We would like to know the velocity of the block when it reaches some position x. Finding this requires an integration. However, acceleration is defined as a derivative with respect to time, which leads to integrals with respect to time, but the force is given as a function of position. To get around this, use the chain rule to find an alternative definition for the acceleration ax that can be written in terms of vx and dx/ dx.

Answer 1

Answer:

An alternative definition for the acceleration ax that can be written in terms of $v_x$ and $\frac{dv_x}{dx}$ is $a_x=v_x \frac{dv_x}{dx}$

Step-by-step explanation:

We know that :

$a_x=\frac{dv_x}{dt}$

Now we are supposed to find an alternative definition for the acceleration ax that can be written in terms of $v_x$ and $\frac{dv_x}{dx}$

So, We will use chain rule over here :

$a_x=\frac{dv_x}{dt}\\a_x=\frac{dv_x}{dt} \times \frac{dx}{dx}\\a_x=\frac{dv_x}{dx} \times \frac{dx}{dt}\\a_x=\frac{dv_x}{dx} \times \frac{dx}{dt} [\frac{dx}{dt}=v_x]\\a_x=\frac{dv_x}{dx} \times v_x\\a_x=v_x\frac{dv_x}{dx}$

Hence an alternative definition for the acceleration ax that can be written in terms of $v_x$ and $\frac{dv_x}{dx}$ is $a_x=v_x \frac{dv_x}{dx}$