Question

The position of a dragonfly that is flying parallel to the ground is given as a function of time by r⃗=[2.90m+(0.0900m/s2)t2]i^− (0.0150m/s3)t3j^. At what value of t does the velocity vector of the insect make an angle of 360 degree clockwise from the x-axis? What is the value of t?

Answer 1

The value of t does not affect the angle between the velocity vector and the x-axis.

Step-by-step explanation:

To find the value of t at which the velocity vector of the insect makes an angle of 360 degrees clockwise from the x-axis, we need to analyze the velocity vector.

1. First, let's calculate the magnitude of the velocity vector using the given position function. The velocity vector is the derivative of the position vector with respect to time:

|v| = |dr/dt|

1. Using the given position function, we differentiate each component:

|v| = sqrt((2 * 0.0900 * t)^2 + (3 * 0.0150 * t)^2)

1. Simplifying the equation:

|v| = sqrt(0.036 * t2 + 0.0135 * t2)

1. Combining like terms:

|v| = sqrt(0.0495 * t2)

1. Taking the square root and simplifying further:

|v| = 0.222 * t

1. Now, we need to find the value of t when the angle between the velocity vector and the x-axis is 360 degrees.

The angle between two vectors is given by:

cosθ = (v·i)/(|v||i|)

1. Substituting the values:

cos(360) = (0.222 * t)/(|v| * 1)

1. Simplifying:

cos(360) = (0.222 * t)/(0.222 * t)

1. Canceling out the common term:

cos(360) = 1

1. Since the cosine of 360 degrees is equal to 1, we can conclude that the angle between the velocity vector and the x-axis is 360 degrees for any value of t.

Hence, the value of t doesn't affect the angle between the velocity vector and the x-axis.