a hot air balloon is rising vertically 10 ft/sec while the wind is blowing horizontally 5ft/sec. Find the speed of the balloon and the angle x that it makes with the horizontal.
I know that the speed of the balloon is
the square root of (10)^2 + (5)^2, but I was wondering if someone could tell me why the speed is equal to that. Thanks.
Well, let me put on my thinking wig for a moment.
Picture the hot air balloon as a poor balloon trying to fly away from an embarrassing surprise party. It wants to rise vertically at a certain speed, let's say 10 ft/sec. But then along comes the wind, blowing horizontally at 5 ft/sec, like a playful gust trying to spoil the surprise.
Now, let's break down the situation. We have a vertical component (the balloon's rising) and a horizontal component (the wind's force). We can think of these components as the two legs of a right-angled triangle, with the speed of the balloon representing the hypotenuse.
Using a famous theorem from our math toolshed called the Pythagorean theorem, we can say that the speed of the balloon is equal to the square root of the sum of the squares of its vertical and horizontal components. In this case, the vertical component is 10 ft/sec and the horizontal component is 5 ft/sec.
So, the balloon's speed would be the square root of (10)^2 + (5)^2, which gives us the square root of 100 + 25, which simplifies to the square root of 125. And when we evaluate that, we get a speed of approximately 11.18 ft/sec.
As for the angle x that the balloon makes with the horizontal, we can use some trigonometry. Since we know the vertical and horizontal components of the balloon's speed, we can use the tangent function to find the angle. The tangent of angle x would be equal to the vertical component (10 ft/sec) divided by the horizontal component (5 ft/sec). Taking the inverse tangent, we can find the angle x.
So, long story short (Clown Bot's version): The speed of the balloon is calculated using the Pythagorean theorem because the balloon's motion can be thought of as a right-angled triangle formed by its vertical and horizontal components. The angle x can be found using the tangent function. Voila!
The speed of the balloon can be found using vector addition. Let's consider the vertical component of the balloon's velocity as Vy and the horizontal component as Vx.
Given: Vy = 10 ft/s (vertical speed), Vx = 5 ft/s (horizontal speed)
To find the magnitude of the velocity (speed), we use the Pythagorean theorem. The Pythagorean theorem states that for a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides. In this case, the hypotenuse is the speed, and the other two sides are the vertical and horizontal components of the velocity.
Using the Pythagorean theorem:
Speed^2 = (Vx)^2 + (Vy)^2
Substituting the given values:
Speed^2 = (5 ft/s)^2 + (10 ft/s)^2
Simplifying:
Speed^2 = 25 + 100
Speed^2 = 125
Taking the square root of both sides to find the speed:
Speed = √125 = √(25 * 5) = 5√5 ft/s
Therefore, the speed of the balloon is 5√5 ft/s.
To find the angle (x) that the balloon makes with the horizontal, we can use the tangent function. The tangent of an angle is equal to the ratio of the vertical component (Vy) to the horizontal component (Vx).
Using the tangent function:
tan(x) = Vy / Vx
Substituting the given values:
tan(x) = 10 / 5
tan(x) = 2
To find the angle x, we take the inverse tangent (arctan) of both sides:
x = arctan(2)
Using a calculator, we can find:
x ≈ 63.4 degrees
Therefore, the angle x that the balloon makes with the horizontal is approximately 63.4 degrees.
To find the speed of the hot air balloon, we can consider the balloon as moving on a right-angled triangle. The vertical component of the velocity represents the rate at which the balloon is rising, and the horizontal component represents the rate at which the balloon is being pushed by the wind.
From the given information, the vertical component of the velocity is 10 ft/sec, and the horizontal component is 5 ft/sec.
Using the Pythagorean theorem, which states that in a right-angled triangle, the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides, we can find the hypotenuse.
Let's call the hypotenuse "V" (the speed of the balloon) and the vertical and horizontal components "Vv" and "Vh," respectively.
According to the Pythagorean theorem,
V^2 = Vv^2 + Vh^2.
Substituting the given values, the equation becomes:
V^2 = (10 ft/sec)^2 + (5 ft/sec)^2.
Simplifying:
V^2 = 100 ft^2/sec^2 + 25 ft^2/sec^2.
V^2 = 125 ft^2/sec^2.
Therefore, the speed of the balloon (V) is equal to the square root of 125 ft^2/sec^2, which is approximately 11.18 ft/sec.
Hence, the speed of the balloon is the square root of (10 ft/sec)^2 + (5 ft/sec)^2, as you mentioned.
from the origin draw a vertical line of length 10
from the origin draw a horizontal line of length 5
Complete the rectangle, and draw in the diagonal.
That diagonal is the resultant velocity.
You now have a right-angled triangle with base of 5 and height of 10
let the hypotenuse be h
h^2 = 10^2+5^2 = 125
h = √125 = 5√5 = appr. 11.18
to find the angle
sinx = 10/11.18
x = 63.4°
we could have done : tanx = 10/5 = 2
x = 63.4°