To solve the problem of the stone dropped from a balloon ascending at a uniform velocity, we need to consider both the initial conditions and the motion of the stone under the influence of gravity. Here's how we can break it down step-by-step.
Setting the Scene
When the stone is released, it starts with an initial upward velocity equal to that of the balloon, which is 5.0 m/s. The balloon is at a height of 50 m above the ground when the stone is released. Meanwhile, the stone will experience a downward acceleration due to gravity, denoted as g, which is 10 m/s².
Understanding the Motion of the Stone
The stone's motion can be described using the kinematic equation for displacement:
s = ut + (1/2)at²
- s = displacement from the initial position (the height of the balloon)
- u = initial velocity of the stone (5.0 m/s upwards)
- a = acceleration due to gravity (–10 m/s², negative because it's directed downward)
- t = time taken for the stone to hit the ground
Calculating Time to Hit the Ground
Since the stone is thrown upwards, it will first ascend before descending. The total height from which it falls is the initial height of the balloon plus the height it travels upwards before it starts falling back down.
Step 1: Find the time to reach the maximum height.
At maximum height, the final velocity is 0 m/s. We can use the equation:
v = u + at
Setting v = 0, we can rearrange to find time (t) to reach the maximum height:
0 = 5.0 m/s - (10 m/s²) * t
10t = 5
t = 0.5 s
Step 2: Calculate the maximum height.
Now we can find out how high the stone goes above the initial height (50 m) using the displacement formula:
s = ut + (1/2)at²
s = (5.0 m/s)(0.5 s) + (1/2)(-10 m/s²)(0.5 s)²
s = 2.5 m - 1.25 m = 1.25 m
The maximum height of the stone above the ground is:
50 m + 1.25 m = 51.25 m
Calculating the Fall to the Ground
After reaching the maximum height, the stone will fall from 51.25 m to the ground. We now need to calculate the time taken to fall this distance using the same kinematic equation:
s = ut + (1/2)at²
Here, the initial velocity (u) is 0 (as it starts falling) and we need to find t when s = 51.25 m:
51.25 = 0*t + (1/2)(10)t²
51.25 = 5t²
t² = 10.25
t = sqrt(10.25) ≈ 3.2 s
Final Height Calculation
During the entire time the stone is in motion, the balloon continues to ascend at 5 m/s. The total time the stone is in the air is approximately 0.5 s (upwards) + 3.2 s (downwards) = 3.7 s.
Therefore, the height of the balloon when the stone hits the ground is:
Height = Initial Height + (Velocity * Time)
Height = 50 m + (5 m/s * 3.7 s) = 50 m + 18.5 m = 68.5 m
Final Answer
The height of the balloon when the stone hits the ground is approximately 68.5 m. Since this value rounds to 68 m, the correct answer is (C) 68.
