The displacement of a dragonfly that is flying parallel to the ground is given as a function of time by r = (2.9 + 0.09*t^2)*i - 0.015*t^3*j.
A. At what value t does the velocity vector of the insect make an angle of 36.0 degrees clockwise from the x-axis?
B. At the time calculated in part (A), what is the magnitude of the acceleration vector of the insect?
C. At the time calculated in part (A), what is the direction of the acceleration vector of the insect?
The displacement of a dragonfly that is flying parallel to the ground is given as a function of time by: r = (2.90 + 0.09*t^2)i - (0.015)t^3j.
The velocity in the x-axis is drx/dt = 2(0.09)t = 0.18t
The velocity in the y-axis is dry/dt = (-0.015*3)t^2= -0.045t^2
The time t at which the velocity vector of the insect makes an angle of 36 degrees clockwise with the x-axis has to be determined.
At t, 0.045t^2/0.18t = tan 36
=> t = 2.906 s
The acceleration in the x-axis is d^2rx/dt^2 = 2*0.09 and the acceleration in the y-axis is -2*0.045t = 0.09t
At t = 2.906 s, the x-component is 0.18 and the y-component is -0.2615
The magnitude of the acceleration is sqrt(0.18^2 + 0.2615^2) = 0.31746 m/s.
The direction of acceleration is arc tan( 0.2615/0.18) = 55.45 clockwise to the x-axis.