A stone is thrown down a well at a velocity of 20m/s. If the well is 20m deep, calculate the time taken for the stone to reach the bottom of the well.
To calculate the time taken for the stone to reach the bottom of the well, we can utilize the equation of motion for vertical free fall:
s = ut + (1/2)gt^2
Where:
s = displacement (in this case, the depth of the well = 20m)
u = initial velocity (20m/s)
t = time taken
g = acceleration due to gravity (approximately 9.8m/s^2)
Plugging in the given values into the equation, we have:
20 = (20)t + (1/2)(9.8)(t^2)
Rearranging the equation, we get:
(1/2)(9.8)(t^2) + (20)t - 20 = 0
To solve this quadratic equation for t, we can use the quadratic formula:
t = (-b ± √(b^2 - 4ac))/(2a)
In this case, a = (1/2)(9.8), b = 20, and c = -20.
Substituting the values into the quadratic formula, we get:
t = (-(20) ± √((20)^2 - 4(1/2)(9.8)(-20)))/(2(1/2)(9.8))
Simplifying further, we have:
t = (-20 ± √(400 + 784))/9.8
t = (-20 ± √1184)/9.8
t = (-20 ± √32 * √37)/9.8
t = (-20 ± 5.657)/9.8
Now we can calculate the two possible values for t:
t₁ = (-20 + 5.657)/9.8 ≈ 0.548s
t₂ = (-20 - 5.657)/9.8 ≈ -2.246s
Since time cannot be negative, we discard the negative value.
Therefore, the time taken for the stone to reach the bottom of the well is approximately 0.548 seconds.
To calculate the time taken for the stone to reach the bottom of the well, you can use the equation of motion for vertically downward motion:
s = ut + (1/2)gt^2
Where:
s = depth of the well = -20m (since the stone is moving downward)
u = initial velocity = -20m/s (since the stone is thrown downward)
g = acceleration due to gravity = -9.8m/s^2 (negative because it acts in the opposite direction of the initial velocity)
t = time taken (what we're trying to find)
Plugging in the given values into the equation, we have:
-20 = (-20)t + (1/2)(-9.8)t^2
Simplifying the equation:
-20 = -20t - 4.9t^2
Rearranging the equation:
4.9t^2 - 20t - 20 = 0
This is a quadratic equation in terms of t. We can solve it using the quadratic formula:
t = (-b ± √(b^2 - 4ac)) / (2a)
In this case, a = 4.9, b = -20, and c = -20. Plugging in these values, we get:
t = (20 ± √((-20)^2 - 4 * 4.9 * -20)) / (2 * 4.9)
Simplifying further:
t = (20 ± √(400 + 392)) / 9.8
t = (20 ± √(792)) / 9.8
Now, we have two possible solutions for t because of the ± sign. Taking the positive square root:
t = (20 + √792) / 9.8
Calculating t using a calculator:
t ≈ 2.204s
Therefore, it will take approximately 2.204 seconds for the stone to reach the bottom of the well.
depth= vi*t-4.9t^2 where vi=-20, d=-20
solve for t.