Suppose the speed of sound is c, and gravity is g. If you throw a rock from an initial height h0 and with an initial velocity v0 the height h(t) of the rock after time t is
h(t)=−g2t^2+v0t+h0.
(The height is negative when the rock is below ground level. Thus you can think of depth as negative height.)
Suppose you throw the rock down a well and hear the impact after t seconds. Then the depth of this well is
d= ????.
(Your answer will be a mathematical expression involving t, g, and c.)
Set up and solve a quadratic equation.
First of all, your equation contains a typo, it should say:
h(t)= (−g/2) t^2+v0t+h0
secondly, what units are you using?
thirdly, did you drop the rock or did you give it a downwards velocity?
let me know before I can answer your question.
Just noticed that Steve answered your question yesterday.
http://www.jiskha.com/display.cgi?id=1468295145
Why are you repeating it? (this time with a typo)
thx could not find it
To find the depth of the well, we can set up a quadratic equation using the given information.
We know that the height of the rock after time t is given by the equation:
h(t) = -g/2 * t^2 + v0 * t + h0
Since the height is negative when the rock is below ground level, the depth of the well can be represented as -h(t). Therefore, the depth of the well, d, is:
d = -h(t)
To find the value of h(t), we substitute t into the equation. Since we hear the impact of the rock after t seconds, we can consider this as the time when h(t) becomes zero.
Setting h(t) to zero, we have:
0 = -g/2 * t^2 + v0 * t + h0
Now we have a quadratic equation in terms of t. To find the depth, we need to solve for t. We can do this by using the quadratic formula:
t = (-b ± √(b^2 - 4ac)) / (2a)
In this equation, a = -g/2, b = v0, and c = h0.
Substituting these values into the quadratic formula, we have:
t = (-(v0) ± √((v0)^2 - 4*(-g/2)*h0)) / (2*(-g/2))
Simplifying further, we get:
t = (-v0 ± √(v0^2 + 2gh0)) / -g
Therefore, the depth of the well, d, can be written as:
d = -h(t) = -(-g/2 * t^2 + v0 * t + h0)
d = g/2 * t^2 - v0 * t - h0
Substitute the value of t from the quadratic formula into this expression to get the final mathematical expression for the depth of the well, in terms of t, g, and c.