A particle moves along a straight line with its position at time \( t \) given by \( s(t) = t^3 - 6t^2 + 9t \). Find the velocity of the particle at \( t = 4 \).

["Title: Find the Velocity of a Particle at ( t = 4 ) Using Motion Equation ( s(t) = t^3 - 6t^2 + 9t )", "When studying motion along a straight line, one of the most fundamental quantities to analyze is velocity, which describes how fast the position of a particle changes over time. In this article, we’ll determine the velocity at ( t = 4 ) for a particle moving according to the position function:", "[\ns(t) = t^3 - 6t^2 + 9t\n]", "---", "### Understanding Position, Velocity, and Their Relationship", "The position function ( s(t) ) gives the location of the particle at time ( t ). The velocity ( v(t) ), however, is the rate of change of position with respect to time, mathematically defined as the first derivative of position:", "[\nv(t) = \frac{ds}{dt}\n]", "By computing this derivative, we obtain the instantaneous velocity at any time ( t ). This allows us to analyze motion trends, such as whether the particle is speeding up or slowing down.", "---", "### Step 1: Differentiate the Position Function", "Given:\n[\ns(t) = t^3 - 6t^2 + 9t\n]", "We differentiate term by term using standard rules of differentiation:", "- Derivative of ( t^3 ) is ( 3t^2 )\n- Derivative of ( -6t^2 ) is ( -12t )\n- Derivative of ( 9t ) is ( 9 )", "So,\n[\nv(t) = \frac{ds}{dt} = 3t^2 - 12t + 9\n]", "---", "### Step 2: Evaluate Velocity at ( t = 4 )", "Now, substitute ( t = 4 ) into the velocity function:", "[\nv(4) = 3(4)^2 - 12(4) + 9\n]", "Calculate step by step:", "- ( 4^2 = 16 )\n- ( 3 \ imes 16 = 48 )\n- ( 12 \ imes 4 = 48 )\n- So,\n[\nv(4) = 48 - 48 + 9 = 9\n]", "---", "### Step 3: Interpretation and Summary", "At time ( t = 4 ), the particle’s velocity is ( v = 9 ) units per time (e.g., meters per second or similar units, depending on ( t )'s units). This positive velocity indicates the particle is moving forward (assuming positive direction corresponds to forward motion) along the line.", "---", "### Final Answer", "The velocity of the particle at ( t = 4 ) is\n[\n\boxed{9}\n]\nunits per unit time.", "---", "### Bonus: Graphing Insight", "Plotting ( v(t) = 3t^2 - 12t + 9 ) reveals a parabola opening upward with roots at ( t = 1 ) and ( t = 3 ), confirming velocity decreases from ( t = 1 ) to ( t = 3 ) before increasing again. At ( t = 4 ), velocity remains positive and growing—consistent with our calculation.", "---", "Keywords: particle motion, velocity, position function, calculus, differentiation, velocity equation, ( s(t) = t^3 - 6t^2 + 9t ), ( v(t) ), ( t = 4 ), physics problems", "Optimized for search engines with clear headings, technical accuracy, and practical explanation."]