A stone is dropped into a river from a bridge 45.4 m above the water. Another stone is thrown vertically down 1.14 s after the first is dropped. Both stones strike the water at the same time. What is the initial speed of the second stone?
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To find the initial speed of the second stone, we can use the equation of motion for free fall:
h = (1/2) * g * t^2
Where:
- h is the height (in this case, it is 45.4 m)
- g is the acceleration due to gravity (approximately 9.8 m/s^2)
- t is the time (in this case, it is 1.14 s)
Let's calculate the time taken for the first stone to reach the water using the given height:
45.4 = (1/2) * 9.8 * t^2
Rearranging the equation to solve for t:
t^2 = (2 * 45.4) / 9.8
t^2 = 92.8 / 9.8
t^2 ≈ 9.47
Taking the square root of both sides:
t ≈ √9.47
t ≈ 3.08 s
Now, let's calculate the initial speed of the second stone using the time (t - t1) it takes after the first stone is dropped:
h = (1/2) * g * (t - t1)^2
Where:
- h is the height (in this case, it is 45.4 m)
- g is the acceleration due to gravity (approximately 9.8 m/s^2)
- t is the time (in this case, it is t = 1.14 s)
- t1 is the time when the first stone is dropped (t1 ≈ 3.08 s)
We already know the height and the time, so let's solve for the initial speed (v0):
45.4 = (1/2) * 9.8 * (1.14 - 3.08)^2 + v0 * (1.14 - 3.08)
Since the second stone is thrown vertically downward, its initial speed (v0) would be negative. Rearranging the equation to solve for v0:
v0 = ((45.4 - (1/2) * 9.8 * (1.14 - 3.08)^2)) / (1.14 - 3.08)
Calculating the values within the parentheses:
v0 = (45.4 - (1/2) * 9.8 * (-1.94)^2) / (-1.94)
Calculating the values within the parentheses and dividing:
v0 ≈ (45.4 - 18.928) / (-1.94)
v0 ≈ -26.472 / (-1.94)
v0 ≈ 13.656 m/s
Therefore, the initial speed of the second stone is approximately 13.656 m/s, given the provided information.