Q A particle is moving in straight line. The velocity v of the particle varies with time t as v

= t²-4t, then the distance traveled by the particle in t = 0s to t = 6s

a) 64/3 b)0 c)32/3 d)none

For 5 points give the steps as well.

To determine the distance traveled by the particle described by the equation \( s(t) = t^2 - 4t \) from \( t = 0 \) seconds to \( t = 6 \) seconds, we need to analyze the motion of the particle over this time interval. The first step is to find the position of the particle at the specified times and then calculate the total distance traveled.

Step-by-Step Calculation

1. Find the Position Function

The position function is given as:

s(t) = t^2 - 4t

2. Calculate the Position at Key Points

We will evaluate the position function at \( t = 0 \) and \( t = 6 \), as well as find any critical points within this interval where the particle might change direction.

• At \( t = 0 \):

s(0) = 0^2 - 4(0) = 0

• At \( t = 6 \):

s(6) = 6^2 - 4(6) = 36 - 24 = 12

3. Find Critical Points

To find critical points, we take the derivative of the position function:

s'(t) = 2t - 4

Setting the derivative equal to zero to find when the velocity is zero:

2t - 4 = 0

t = 2

4. Evaluate the Position at the Critical Point

Now, we calculate the position at \( t = 2 \):

s(2) = 2^2 - 4(2) = 4 - 8 = -4

5. Determine the Total Distance Traveled

Now we have the positions at the critical points and endpoints:

• s(0) = 0
• s(2) = -4
• s(6) = 12

To find the total distance traveled, we consider the absolute changes in position:

• From \( t = 0 \) to \( t = 2 \): Distance = |s(2) - s(0)| = |-4 - 0| = 4
• From \( t = 2 \) to \( t = 6 \): Distance = |s(6) - s(2)| = |12 - (-4)| = |12 + 4| = 16

Adding these distances together gives us:

Total Distance = 4 + 16 = 20

Final Answer

Since the total distance traveled by the particle from \( t = 0 \) to \( t = 6 \) seconds is 20, which is not one of the provided options (64/3, 0, 32/3), the correct choice is d) none.