Conservation of energy as a function of Time

[PlantPage55]

Homework Statement

A ball of unspecified mass is in a free fall - and we are supposed to rearrange equations to teach us the basics of kinematics. It's a general question with no values given for the variables

Use your conservation of energy equation: v = sqrt(2gh) and calculus to write an expression for the ball's position as a function of time.

Homework Equations

It's the last part that is getting me. I'm thinking they want me to use an integral to find the position-time graph or something. Alternatively, I can't figure out how to relate a kinematic equation: y=(1/2)gt^2 (in this case) to the conservation of energy equation we derived.

(We derived this conservation of energy equation from putting the kinetic energy in terms of its velocity - then relating this energy transfer to the force of gravity working on this free falling ball. The result was the above equation - which is supposed to be an equation of velocity in terms of the forces acting on the ball. Did I do that right?


3. Attempt at Solution

I've been bothered by this for hours. I tried writing the integral of the v = sqrt(2gh) function, but I just get a messy integral and I'm not sure that's the right idea anyway. Maybe I'm misinterpreting the question?

Thanks for any help you can give!

 

[nicktacik]

Okay let's write v = sqrt(2gh), in calculus terms

$$\frac{dy}{dt} = \sqrt{2gy(t)$$

Or

$$\frac{dy}{\sqrt{2gy(t)}} = dt$$

Or if you integrate both sides

$$\int{\frac{dy}{\sqrt{2gy(t)}}} = \int{dt} = t + C$$This is an easy enough integral, then solve for y(t) and use your initial conditions to find C.

 

[andrevdh]

Try using

$$v = \frac{dy}{dt}$$

and

$$y(t)=h_o - h(t)$$

y positive downwards.

 

[PlantPage55]

Excellent! Thank you - that makes great sense!