The velocity vector of a sprinting cheetah has x- and y-components vx = +12.6 m/s and vy = -27.2 m/s. What angle does the velocity vector make with the +x- and −y-axes?
Ah, the sprinting cheetah! Such a majestic creature. Now, let's figure out that angle.
First, let's break out our trusty trigonometry skills. We can use the tangent function to find the angle.
So, the angle θ between the velocity vector and the +x-axis can be found using the formula:
θ = tan^-1(vy/vx)
Plugging in the values we have, we get:
θ = tan^-1((-27.2 m/s) / (12.6 m/s))
Now, let's calculate that angle using my handy-dandy calculator:
θ ≈ -63.2 degrees
Ah, but we mustn't forget about the -y-axis angle as well. We simply take the negative of the previous angle, which gives us:
-θ ≈ 63.2 degrees
So, the angle that the velocity vector makes with the +x-axis is approximately -63.2 degrees, and with the -y-axis, it is 63.2 degrees. Quite a fancy footwork by our sprinting cheetah, wouldn't you say?
To find the angle that the velocity vector makes with the +x-axis, you can use the formula:
θ = arctan(vy/vx)
where θ is the angle and vx and vy are the x- and y-components of the velocity vector, respectively.
Plugging in the values:
θ = arctan((-27.2)/(12.6))
Using a calculator:
θ ≈ -66.8 degrees
Therefore, the angle that the velocity vector makes with the +x-axis is approximately -66.8 degrees.
To find the angle that the velocity vector makes with the +x- and -y-axes, we can use trigonometry.
First, let's find the angle with the +x-axis. We can use the inverse tangent function (tan^(-1)) to find the angle.
The formula to find the angle (θ) is:
θ = tan^(-1)(vy / vx)
Plugging in the given values:
θ = tan^(-1)(-27.2 m/s / 12.6 m/s)
Using a scientific calculator or a math software, we can find the arctan(-27.2 / 12.6) = -66.54 degrees (rounded to two decimal places).
So, the angle that the velocity vector makes with the +x-axis is approximately -66.54 degrees.
Now, let's find the angle with the -y-axis. Since the -y-axis is in the opposite direction of the +y-axis, the angle will be the supplement of the angle made with the +y-axis.
The angle with the +y-axis can be found using the formula:
θ = tan^(-1)(vx / vy)
Plugging in the given values:
θ = tan^(-1)(12.6 m/s / -27.2 m/s)
Using the same method, we find that the angle with the +y-axis is approximately -24.74 degrees.
The angle with the -y-axis will be the supplement of this angle:
Angle with -y-axis = 180 degrees - (-24.74 degrees) = 204.74 degrees.
Therefore, the angle that the velocity vector makes with the +x-axis is approximately -66.54 degrees, and the angle with the -y-axis is approximately 204.74 degrees.