A hot hair balloon is rising vertically 20 ft/sec while the wind is blowing horizontally at 15 ft/sec. Find the speed v of the balloon and the angle x it makes with the horizontal.
I know how to solve this problem, but I don't understand why or how you know it's a right angle.
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y axis velocity component = 20
x axis component = 15
tan x = 20/15
x = tan^-1 (1.333) = 53.1 degrees
To solve this problem, we need to use vector addition. The balloon has two velocity components: vertical velocity (upward direction) and horizontal velocity (due to wind).
Let's denote the vertical velocity as Vy and the horizontal velocity as Vx. We know that Vy = 20 ft/sec (the balloon is rising vertically at a rate of 20 ft/sec) and Vx = 15 ft/sec (the wind is blowing horizontally at 15 ft/sec).
To find the speed v of the balloon, we need to find the magnitude of the resultant velocity vector. Using Pythagoras' theorem, we can calculate the magnitude:
v = sqrt(Vx^2 + Vy^2)
v = sqrt((15 ft/sec)^2 + (20 ft/sec)^2)
v = sqrt(225 ft^2/sec^2 + 400 ft^2/sec^2)
v = sqrt(625 ft^2/sec^2)
v = 25 ft/sec
So, the speed of the balloon is 25 ft/sec.
To find the angle x that the balloon makes with the horizontal, we can use trigonometry. We can find the tangent of the angle:
tan(x) = Vy / Vx
tan(x) = 20 ft/sec / 15 ft/sec
tan(x) = (20/15)
tan(x) = 4/3
Now, we can find the angle x:
x = arctan(4/3)
x ā 53.13 degrees
So, the angle that the balloon makes with the horizontal is approximately 53.13 degrees. Note that this angle is not a right angle.
To solve this problem, we can use vector addition because we have two velocities acting in different directions. Let's start by labeling the given information:
Vertical velocity of the balloon (upwards) = 20 ft/sec
Horizontal velocity due to the wind (rightwards) = 15 ft/sec
To find the speed v of the balloon, we need to find the magnitude of the resultant vector that combines the vertical and horizontal velocities. We can calculate this using the Pythagorean theorem:
v = sqrt((vertical velocity)^2 + (horizontal velocity)^2)
= sqrt((20 ft/sec)^2 + (15 ft/sec)^2)
= sqrt(400 ft^2/sec^2 + 225 ft^2/sec^2)
= sqrt(625 ft^2/sec^2)
= 25 ft/sec
Therefore, the speed of the balloon is 25 ft/sec.
Now let's find the angle x that the balloon makes with the horizontal. We can use the tangent function to find this angle:
tan(x) = (vertical velocity)/(horizontal velocity)
= (20 ft/sec) / (15 ft/sec)
= (4/3)
To find the angle x, we can take the inverse tangent (arctan) of this ratio:
x = arctan(4/3)
ā 53.13 degrees
So, the angle x that the balloon makes with the horizontal is approximately 53.13 degrees.
To address your question about the right angle, in this problem, there is no mention of a right angle. The angles involved are the angle of the balloon's motion with respect to the horizontal. The reason we can use the tangent function to find this angle is due to the fact that the balloon's motion is independent of the wind's motion.