To solve this problem, we need to analyze the motion of the insect and the rod separately, considering both linear and angular motion. The insect moves along the rod while the rod itself is rotating. Let's break down the components step by step.

Understanding the Motion of the Insect

The insect moves with a constant velocity of 3 m/s along the rod. Since it starts from the origin and moves in the positive y-direction, after 2 seconds, its position vector can be calculated as follows:

• Velocity of the insect, v = 3 m/s
• Time, t = 2 s
• Position vector after 2 seconds: r = v × t = 3 m/s × 2 s = 6 m

Since the rod is initially along the x-axis and the insect moves in the positive y-direction, the position vector of the insect at t = 2 seconds is:

Position vector = 6j

Calculating the Speed of the Insect

Next, we need to determine the speed of the insect relative to an observer on the ground. The rod is rotating with a constant angular acceleration. Given that the angular acceleration is constant, we can find the angular velocity after 2 seconds:

• Let the angular acceleration be α (in rad/s²).
• Since we don't have a specific value for α, we can denote it as α.
• Angular velocity after 2 seconds: ω = α × t = α × 2

Now, the speed of the insect relative to the ground can be calculated by considering both its linear speed along the rod and the additional speed due to the rotation of the rod. The insect is at a distance of 6 m from the origin (the center of rotation) after 2 seconds. The tangential speed due to the rod's rotation is given by:

v_t = r × ω = 6 m × (α × 2)

Thus, the total speed of the insect relative to the ground is:

Total speed = Linear speed + Tangential speed = 3 m/s + 6 m × (α × 2)

Determining Tangential Acceleration

The tangential acceleration of the insect is directly related to the angular acceleration of the rod. The tangential acceleration (a_t) can be calculated using the formula:

a_t = r × α

Substituting the distance (r = 6 m) into the equation gives:

a_t = 6 m × α

Summary of Results

To summarize the findings:

• Position vector of the insect after 2 seconds: 6j
• Speed of the insect relative to the ground: 3 m/s + 12α m/s
• Tangential acceleration of the insect: 6α m/s²

In this analysis, we have considered both the linear motion of the insect and the rotational motion of the rod. The results depend on the angular acceleration of the rod, which was not specified in the problem. If you have a specific value for α, you can substitute it into the equations to find numerical answers for speed and tangential acceleration.