The altitude of a hang glider is increasing at a rate of 7.35 m/s. At the same time, the shadow of the glider moves along the ground at a speed of 14.5 m/s when the sun is directly overhead. Find the magnitude of the glider's velocity.
v=sqrt[v(x)²+v(y)²]=
=sqrt[7.35²+14.5²]=16.26 m/s
Why did the hang glider bring an umbrella with him? Because he wanted to stay in the shade!
To find the magnitude of the glider's velocity, we can use the Pythagorean theorem. The glider's velocity can be represented by a right triangle, where one side is the rate of increase in altitude (7.35 m/s) and the other side is the shadow's speed along the ground (14.5 m/s).
By using the Pythagorean theorem (a^2 + b^2 = c^2), we can calculate the magnitude of the glider's velocity:
velocity^2 = (altitude rate)^2 + (shadow speed)^2
velocity^2 = (7.35 m/s)^2 + (14.5 m/s)^2
Now it's time for some math magic! Crunching the numbers gives us:
velocity^2 = 53.9225 + 210.25
velocity^2 = 264.1725
Taking the square root of both sides, we find:
velocity ≈ √264.1725
velocity ≈ 16.258 m/s
So, the magnitude of the glider's velocity is approximately 16.258 m/s. Keep soaring high and staying cool in that shade!
To find the magnitude of the glider's velocity, we need to use vector addition.
Let's call the velocity of the glider V and the velocity of the shadow S. We know that the vertical component of the glider's velocity is given by Vv = 7.35 m/s (since the altitude is increasing at this rate), and the horizontal component of the shadow's velocity is given by Sh = 14.5 m/s (since the shadow is moving along the ground at this speed).
Since the sun is directly overhead, the shadow's velocity is purely horizontal, so S = Sh î (where î is the unit vector in the horizontal direction). The velocity of the glider can be represented as the sum of its horizontal and vertical components: V = Vh î + Vv ĵ (where ĵ is the unit vector in the vertical direction).
Now, to find Vh (the horizontal component of the glider's velocity), we can use the fact that the magnitude of the glider's velocity is equal to the magnitude of the shadow's velocity. Therefore:
|V| = |S|
|Vh î + Vv ĵ| = |Sh î|
√(Vh^2 + Vv^2) = Sh
Substituting the given values, we have:
√(Vh^2 + (7.35 m/s)^2) = 14.5 m/s
Squaring both sides of the equation:
Vh^2 + (7.35 m/s)^2 = (14.5 m/s)^2
Vh^2 = (14.5 m/s)^2 - (7.35 m/s)^2
Vh^2 = 210.25 m^2/s^2 - 54.0225 m^2/s^2
Vh^2 = 156.2275 m^2/s^2
Taking the square root of both sides:
Vh = √(156.2275 m^2/s^2)
Vh ≈ 12.5 m/s
So, the magnitude of the glider's velocity is approximately 12.5 m/s.