Ugh. Dude, please tell me you aren't actually a physics teacher. This is super cringey.
Now the SUVAT equation we will use is S = UT + 1/2AT SQUARED (I can't do superscript from my phone)
Let's rearrange to get time...?
This gives
T squared = S/(UT +1/2A)
Algebra is wrong.
T
2 = 2(S - UT)/A.
Doesn't matter though. Read on.
Here S = distance
T = time
A = acceleration
U = initial starting speed of the earth (not ball, which is where you cocked up)
Be careful defining your terms. Make sure you get the directions consistent.
S = distance between the earth and the ball. S is positive when the ball is ahead of the earth. (S
ball - S
earth)
V = velocity between the earth and the ball. V is positive when the ball is faster than the earth. (V
ball - V
earth)
A = acceleration between the earth and the ball. A is positive when the ball is accelerating faster than the earth. (A
ball - A
earth)
U = initial velocity between earth and ball.
Initially positive and greater than zero, because the ball is moving faster than the earth.You can't immediately solve for T, because you don't know U yet. First solve for U, which happens to be 8.82 m/s by sheer luck. (Symmetry, actually.)
Now you can solve for T:
4 = 8.82T - 1/2AT
2Using the quadratic formula gives your original value for T, 0.9 seconds, again due to pure luck. (Hurray for symmetry!)
Let's put that into our equation to see how long it takes the ball to appear to fall...
T squared = S/(UT + 1/2A)
here the new value for U is 8.82, and T (the T in the brackets) is 0.9
No no no no no!!!! You can not arbitrarily decide that each T is different. They are both the same variable. They are the same value. Right now the
relative velocity between the ball and the earth is zero. The distance is negative this time.
S = UT + 1/2AT
2 -4 =
(0)T + 1/2(-9.8)T
2Solving for T gives, once again, 0.9 seconds. The exact same answer as before.
Also, you mentioned air resistance, but didn't include it in your calculations. What gives?