The total travel time along a frictionless path?

[jeebs]

Hi,
I have this problem about an object moving along a track due to a uniform gravitational field. It moves from being at rest in the higher position A (x_A,y_A) to the lower position B (x_B,y_B). There is no friction at all. What I am supposed to be doing is showing that the total time from A to B is given by:

t_t_o_t = \frac{1}{\sqrt{2g}}\int dt \sqrt{\frac{(dx/dt)^2 + (dy/dt)^2}{y}}
where the limits of the integral are given as between (x_A,y_A) and (x_B,y_B).

The limits I do not quite understand, because surely they should really be times rather than positions if we are integrating with respect to time?

Anyway, what I have attempted is this:
From Newton's equation of motion, v² = u² + 2as, if we set initial velocity u = 0, gravitational acceleration a = g and vertical displacement s = y, then we have an expression for the object's final velocity, v = \sqrt{2gy}.

Also, I said that the instantaneous velocity will be given by v_i = \sqrt{(dx/dt)^2 + (dy/dt)^2}.

Since these two things match the denominator and numerator in the integrand respectively, they must be important. Clearly v_i/v = 0 at time t=0 and position A, and increases to v_i/v = 1 at time t=t_tot and position B.


This is where I start to hit trouble though. I'm not really sure what to make of these spatial limits and I can't see the way forward. One thing I have tried is to say that

(\sqrt{(dx/dt)^2 + (dy/dt)^2})dt = \sqrt{dt^2(dx/dt)^2 + dt^2(dy/dt)^2} = \sqrt{(dx)^2 + (dy)^2} = dy\sqrt{(dx/dy) + 1}

but when I proceed with the integral integrating with respect to y from there, I end up with a quantity that does not have dimensions of time.

Can anyone steer me in the right direction?
Many thanks to those who try.

 

[chrisk]

What is the direction of the instantaneous velocity

v_i

and compare with your final velocity direction

v

If the dt outside the square root is moved inside the square root, as you have shown, what are the units of the integrand? Then determine the units of

\frac{1}{\sqrt{2g}}

You should find the units of the expression is time.

 

[Cyosis]

Start out with ds=vdt. From this it follows that t=\int \frac{ds}{v}. You know that ds=\sqrt{dx^2+dy^2} which you can rewrite to something in terms of dt as you have already shown in your last line of equations. Now you just have to calculate v. Hint: use conservation of energy.

 

[jeebs]

Start out with ds=vdt. From this it follows that t=\int \frac{ds}{v}. You know that ds=\sqrt{dx^2+dy^2} which you can rewrite to something in terms of dt as you have already shown in your last line of equations. Now you just have to calculate v. Hint: use conservation of energy.

Thanks guys.

What I've tried now is this:-
In the equation ds = vdt, this velocity is not a final or initial velocity as are used in Newton's equations of motion, it is some instantaneous velocity v=v(t) = v(y).

Also, kinetic energy E_k = 0.5mv² and potential energy E_p = -mgy.

Potential energy is at a maximum of zero when the object is at its highest, and as it falls along its path it becomes more negative. At t=0, velocity is also 0, so that the sum of the kinetic and potential energies is zero, ie. E_k + E_p = 0, which is a constant.

This means that at some downwards displacement y, we have 0.5mv² - mgy = 0 at all times. This leads directly to v=(2gy)^1/2, which then slots into my final expression for t=t_tot.

Am I justified in saying this?