I was wondering how I would go about calculating the time a object needs to travel from point $A(0,0)$ to point $B(1,-1)$ while traveling along the line $-x^{1/2}$.
The object is accelerating in a uniform gravitational field of strength $9.81$m/s and experiences no friction or air resistance.
Thanks for any help.
Best Answer
The vertical speed is
$$\frac{dy}{dt}=v \sin\theta$$
where $v = \sqrt{-2gy}$ (derived from $\frac12mv^2=-mgy $) and $\tan\theta = \frac{dy}{dx} = \frac1{2y}$. Plug them into the above expression to get,
$$\frac{dy}{dt}= \sqrt{\frac{-2gy}{4y^2+1}}$$
Thus, the time is obtained from,
$$T = \int_0^{-1} \sqrt{\frac{4y^2+1}{-2gy}}dy=0.584$$