While standing on a bridge 15.0 m above the ground, you drop a stone from rest. When the stone has fallen 3.10 m, you throw a second stone straight down. What initial velocity must you give the second stone if they are both to reach the ground at the same instant? Take the downward direction to be the negative direction
first get time to fall to ground
a = -9.8
v = -9.8 t
-15 = -4.9 t^2
t = sqrt (15/4.9) = 1.75 seconds
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Now how long did it take to reach 3.1 m down
3.1 = -4.9t^2
t = .795 seconds
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so the second rock spends 1.75 -.795 or .955 seconds in the air
-15 = Vo (.955) -4.9 (.955)^2
.955 Vo = 3.64-15
Vo = - 11.9
so 11.9 m/s downward (negative)
To find the initial velocity of the second stone, we need to determine the time it takes for both stones to reach the ground.
The first stone falls from rest, so we can use the kinematic equation for a falling object:
h = (1/2) * g * t^2
where h is the height, g is the acceleration due to gravity (approximately -9.8 m/s^2), and t is the time.
For the first stone that falls 3.10 m:
3.10 = (1/2) * (-9.8) * t^2
Simplifying the equation:
-4.9t^2 = 3.10
Rearranging the equation:
t^2 = 3.10 / -4.9
Taking the square root of both sides:
t = sqrt(3.10 / -4.9)
Now, let's find the time for the second stone to reach the ground. It starts at a height of 15.0 m, so the equation becomes:
15 = (1/2) * (-9.8) * t^2
Simplifying the equation:
-4.9t^2 = 15
Rearranging the equation:
t^2 = 15 / -4.9
Taking the square root of both sides:
t = sqrt(15 / -4.9)
Since we want both stones to reach the ground at the same instant, we equate the two time values:
sqrt(3.10 / -4.9) = sqrt(15 / -4.9)
Taking the negative square root because the downward direction is negative:
- sqrt(3.10 / 4.9) = sqrt(15 / 4.9)
Simplifying the equation:
- sqrt(3.10 / 4.9) = sqrt(15 / 4.9)
The negative sign indicates that the second stone needs to travel in the opposite direction, which is downward. Therefore, the negative square root is the initial velocity we need to give the second stone.
Thus, the initial velocity of the second stone must be approximately -4.4 m/s.