Question

What is the speed of the particle at t=4 seconds, given that it is moving in a circle of radius 2 meters according to the relation θ = 3t² + 2t

A) 56 m/s
B) 92 m/s
C) 104 m/s
D) 80 m/s

Answer 1

The speed of the particle at (t = 4) seconds is 80 m/s, calculated by substituting the given time value into the expression (v = 2(6t + 2)) derived from the angular position function.Thus,the correct option is d.

Explanation:

The speed of a particle moving in a circular path can be found using the formula
`\(v = (ds)/(dt)\),` where \(s\) is the arc length and (t) is time. In this case,
`\(s = r \cdot \theta\)`, where (r) is the radius and
`\(\theta\)` is the angle in radians.

Given the relation
`\(\theta = 3t^2 + 2t\),` we first find the angular velocity
`\(\omega\)` by taking the derivative of
`\(\theta\)` with respect to
`\(t\): \(\omega = (d\theta)/(dt) = 6t + 2\)`. Now, the speed (v) is given by
`\(v = r \cdot \omega\)`. With the radius (r = 2), we substitute
`\(\omega = 6t + 2\)` into the formula, yielding (v = 2(6t + 2)).

To find the speed at (t = 4) seconds, substitute (t = 4) into the expression:
`\(v = 2(6 \cdot 4 + 2) = 80\)` m/s.

In summary, the speed of the particle at (t = 4) seconds is 80 m/s. This is obtained by first determining the angular velocity from the given angular position function, and then using the formula for linear speed in circular motion. The calculation involves substituting the time value into the expression for speed, resulting in the final answer of 80 m/s.

Therefore,the correct option is d.