Simple problem. Depth of well providing sound reaches you in 1.5 second.

[new324]

I was wondering you someone could help me with an alternative way of solving this problem. Just to help my algebra a bit.
You drop a rock down a well. Find the depth of the well providing the sound of the splash reaches you in 1.5 seconds.

First off, we know speed of sound is 343 m/s.
Here are my equations:
t1+t2=1.5 s t1 is time for rock to hit bottom t2 is time for sound to reach your ears.
x=.5*a*t^2 -> x=4.9*t1^2
v=x/t -> x=343*t2

Now I sub the last 2 equations to get:
343*t2=4.9*t1^2
Now we sub. for t2:
343*(1.5-t1)=4.9*t1^2
From here we can reduce, and solve with the quadratic formula.

I was wondering how you would solve the equation, if you sub. for t1 instead of t2. This would make the equation:
343*t2=4.9*(t2-1.5)^2

 

[Doc Al]

I was wondering how you would solve the equation, if you sub. for t1 instead of t2. This would make the equation:
343*t2=4.9*(t2-1.5)^2

It's just another quadratic equation. Multiply out the term on the right (do the square) and put it into standard form.

 

[M98Ranger]

Thank you for sharing your method for solving this problem. Here is an alternative approach using algebra:

Let d be the depth of the well and t be the time it takes for the sound to reach you. We can use the formula d = rt, where r is the speed of sound, to find the depth.

Since we know the speed of sound is 343 m/s, we can write the equation as d = 343t.

Now, we also know that the total time, t, is equal to the sum of the time it takes for the rock to hit the bottom of the well and the time it takes for the sound to reach you. So we can write the equation as t = t1 + t2.

Since we are given that t = 1.5 seconds and t1 is the time it takes for the rock to hit the bottom, we can write the equation as 1.5 = t1 + t2.

Now, we can substitute this value of t into our first equation to get d = 343(1.5 - t1).

Using the equation x = 4.9t^2, we can also write t1 as √(d/4.9).

Substituting this value into our equation 1.5 = t1 + t2, we get 1.5 = √(d/4.9) + t2.

Solving for t2, we get t2 = 1.5 - √(d/4.9).

Now, we can substitute this value of t2 into our first equation to get d = 343(1.5 - √(d/4.9)).

Simplifying this equation, we get d = 514.5 - 343√(d/4.9).

Solving for d, we get a quadratic equation: 343√(d/4.9) + d = 514.5.

Solving this equation using the quadratic formula, we get two possible solutions: d = 0.45 m or d = 173.68 m.

So, using this method, we can find that the depth of the well is either 0.45 m or 173.68 m.