A stone is dropped from a balloon going up with a uniform velocity of 5.0 m/s. If the balloon was 50 m high when the stone was dropped, find its height (in meters) when the stone hits the ground. Take g = 10 m/s2
(A) 50
(B) 58
(C) 68
(D) 78.5
68
To find the height at which the stone hits the ground, we need to find the time it takes for the stone to fall.
First, let's find the time it takes for the stone to reach the highest point of its trajectory. We can use the equation:
h = ut + (1/2)gt^2
Where:
h = height
u = initial velocity
t = time
g = acceleration due to gravity
At the highest point, the final velocity is zero, so u = 5.0 m/s. The height at this point is 50 m. Plugging in these values, we get:
50 = (5.0)t + (1/2)(10)t^2
Simplifying the equation, we have:
(1/2)(10)t^2 + 5.0t - 50 = 0
To solve this quadratic equation, we can use the quadratic formula:
t = (-b ± sqrt(b^2 - 4ac))/(2a)
In this equation, a = (1/2)(10) = 5, b = 5.0, and c = -50.
Plugging in these values, we get:
t = (-5 ± sqrt(5^2 - 4(5)(-50)))/(2(5))
Simplifying further, we have:
t = (-5 ± sqrt(25 + 1000))/10
t = (-5 ± sqrt(1025))/10
The positive solution will give us the time it takes for the stone to reach the highest point since time cannot be negative.
t = (-5 + sqrt(1025))/10
Using a calculator, we find that t ≈ 3.17 seconds.
Now, we can find the total time it takes for the stone to hit the ground. Since the stone has reached its maximum height after 3.17 seconds, it will take the same amount of time for it to fall back down.
So, the total time to hit the ground is 2 × 3.17 seconds ≈ 6.34 seconds.
Now, let's find the height at this time. We can use the equation:
h = ut + (1/2)gt^2
At this time, the initial velocity is still 5.0 m/s, and the time is 6.34 seconds. Plugging in these values, we get:
h = (5.0)(6.34) + (1/2)(10)(6.34)^2
Simplifying this equation, we have:
h = 31.7 + (1/2)(10)(40.0996)
h = 31.7 + 200.498
h ≈ 231.198
So, the height of the stone when it hits the ground is approximately 231.198 meters.
Therefore, the correct answer is not provided in the options given.