A 124-kg balloon carrying a 22-kg basket is descending with a constant downward velocity of 25.2 m/s A 1.0-kg stone is thrown from the basket with an initial velocity of 11.7 m/s perpendicular to the path of the descending balloon, as measured relative to a person at rest in the basket. The person in the basket sees the stone hit the ground 6.70 s after being thrown. Assume that the balloon continues its downward descent with the same constant speed of 25.2 m/s .
Well, well, well, it seems we have an exciting physics problem on our hands! Let's break it down, shall we?
We have a heavy balloon with a basket, a stone being thrown out of the basket, and a person in the basket watching it all unfold. Quite the circus act!
First, let's calculate the distance the stone traveled horizontally before hitting the ground. We know it took 6.70 seconds to hit the ground, and since the velocity is constant, we can use the equation distance equals velocity multiplied by time. So, horizontal distance equals 11.7 m/s multiplied by 6.70 s.
Horizontal distance = 11.7 m/s x 6.70 s = 78.39 meters. That's quite a throw!
Now, let's focus on the balloon. We know it's descending with a constant velocity of 25.2 m/s. The stone was thrown perpendicular to the path of the balloon, meaning it won't affect the balloon's downward motion. It's like a clown juggling on a unicycle while riding downhill. Quite the balancing act!
So, the vertical distance the balloon traveled during the stone's flight time is 25.2 m/s multiplied by 6.70 s.
Vertical distance = 25.2 m/s x 6.70 s = 168.24 meters. That's quite a descent!
Now, we need to find the total distance the stone traveled. Using the Pythagorean theorem, we can find the hypotenuse of the right triangle formed by the horizontal and vertical distances.
Total distance = square root of (horizontal distance squared plus vertical distance squared).
Total distance = square root of (78.39 meters squared plus 168.24 meters squared).
Calculating this gives us a total distance of approximately 188.55 meters. That stone really flew!
Well, there you have it! The stone traveled approximately 188.55 meters before hitting the ground. Seems like our circus act came to an end, but don't worry, there are always more physics problems to solve under this big top!
To solve this problem, we'll divide it into three main steps:
Step 1: Calculate the time it takes for the stone to hit the ground as observed by the person in the basket.
Step 2: Calculate the horizontal distance traveled by the stone during this time.
Step 3: Calculate the vertical distance traveled by the stone during this time.
Let's start with Step 1:
Step 1: Calculate the time it takes for the stone to hit the ground as observed by the person in the basket.
The stone is thrown with an initial velocity of 11.7 m/s and it travels in a straight line perpendicular to the path of the descending balloon. The person in the basket sees the stone hit the ground after 6.70 s.
Therefore, the time it takes for the stone to hit the ground, as observed by the person in the basket, is 6.70 s.
Step 2: Calculate the horizontal distance traveled by the stone during this time.
Since the stone is thrown perpendicular to the path of the descending balloon, there is no horizontal acceleration acting on it. Therefore, the horizontal velocity of the stone remains constant during its motion.
The horizontal distance traveled by the stone can be calculated using the equation:
distance = velocity × time
The horizontal velocity of the stone is given as 11.7 m/s, and the time taken to hit the ground is 6.70 s.
Therefore, the horizontal distance traveled by the stone is:
distance = 11.7 m/s × 6.70 s = 78.39 m
Step 3: Calculate the vertical distance traveled by the stone during this time.
Since the balloon is descending with a constant velocity of 25.2 m/s, the vertical distance traveled by the stone can be calculated using the equation:
distance = initial velocity × time + 0.5 × acceleration × time^2
Given:
Initial velocity = 0 m/s (since the stone is thrown perpendicular to the path of the balloon)
Acceleration = 9.8 m/s^2 (acceleration due to gravity)
Time = 6.70 s
Substituting the values into the equation, we get:
distance = 0 × 6.70 s + 0.5 × 9.8 m/s^2 × (6.70 s)^2
distance = 0 + 0.5 × 9.8 m/s^2 × 44.89 s^2
distance = 0.5 × 9.8 m/s^2 × 44.89 s^2
distance ≈ 221.87 m
Therefore, the vertical distance traveled by the stone during this time is approximately 221.87 m.
To summarize:
- The time it takes for the stone to hit the ground, as observed by the person in the basket, is 6.70 s.
- The horizontal distance traveled by the stone during this time is 78.39 m.
- The vertical distance traveled by the stone during this time is approximately 221.87 m.
To solve this problem, we can break it down into two parts: the motion of the balloon and the motion of the stone.
First, let's analyze the motion of the balloon. We are given that the balloon is descending with a constant downward velocity of 25.2 m/s. This means that the balloon's vertical speed remains constant throughout its descent. Since the balloon is descending at a constant velocity, we can assume there is no acceleration acting on it in the vertical direction.
Next, let's analyze the motion of the stone. We are given that the stone is thrown from the basket with an initial velocity of 11.7 m/s perpendicular to the path of the descending balloon, as measured relative to a person at rest in the basket. This tells us that the stone is initially moving horizontally with a velocity of 11.7 m/s.
The person in the basket sees the stone hit the ground 6.70 s after being thrown. We can use this information to find the horizontal distance the stone travels. Since the initial horizontal velocity is 11.7 m/s and the time of flight is 6.70 s, we can use the equation:
Distance = Velocity × Time
Distance = 11.7 m/s × 6.70 s = 78.39 meters
Therefore, the stone travels a horizontal distance of 78.39 meters before hitting the ground.
However, we need to find the vertical distance the stone travels. Since the stone is thrown perpendicular to the path of the descending balloon, the stone's vertical motion is not affected by the downward velocity of the balloon. This means that the stone falls freely under the influence of gravity as if it were thrown from rest at an initial height.
To find the vertical distance the stone travels, we can use the equation for free fall motion:
Distance = (1/2) × acceleration × time^2
The acceleration due to gravity is approximately 9.8 m/s^2, and the time of flight is 6.70 s. Plugging these values into the equation, we get:
Distance = (1/2) × 9.8 m/s^2 × (6.70 s)^2 = 229.83 meters
Therefore, the stone travels a vertical distance of 229.83 meters before hitting the ground.
Now, to determine the final position of the stone relative to the balloon, we need to combine the horizontal and vertical distances. Since the stone is initially thrown horizontally and then falls freely, the total displacement can be found using the Pythagorean theorem:
Displacement = √(Horizontal Distance^2 + Vertical Distance^2)
Displacement = √(78.39^2 + 229.83^2) = 246.69 meters
Therefore, the final position of the stone relative to the balloon is approximately 246.69 meters away from the point of release.
It is important to note that the above calculations assume ideal conditions, neglecting factors like air resistance.