A particle is moving in a striaght line and at a given time, its displacement is `S = t^(3) - 6t^(2) + 9t`, where t is in seconds and S is in meter . The total distance travelled by the particle during the first five seconds is

Options

A. 8m
B. 4m
C. 24m
D. 28m

Answer

To find the total distance traveled by the particle during the first five seconds, we first need to analyze the given displacement equation:

\[ S(t) = t^3 - 6t^2 + 9t \]

### Step 1: Calculate Displacement at Different Time Intervals
We will calculate the displacement at \( t = 0, 1, 2, 3, 4, \) and \( 5 \) seconds.

1. **At \( t = 0 \):**
\[ S(0) = 0^3 - 6(0^2) + 9(0) = 0 \text{ meters} \]

2. **At \( t = 1 \):**
\[ S(1) = 1^3 - 6(1^2) + 9(1) = 1 - 6 + 9 = 4 \text{ meters} \]

3. **At \( t = 2 \):**
\[ S(2) = 2^3 - 6(2^2) + 9(2) = 8 - 24 + 18 = 2 \text{ meters} \]

4. **At \( t = 3 \):**
\[ S(3) = 3^3 - 6(3^2) + 9(3) = 27 - 54 + 27 = 0 \text{ meters} \]

5. **At \( t = 4 \):**
\[ S(4) = 4^3 - 6(4^2) + 9(4) = 64 - 96 + 36 = 4 \text{ meters} \]

6. **At \( t = 5 \):**
\[ S(5) = 5^3 - 6(5^2) + 9(5) = 125 - 150 + 45 = 20 \text{ meters} \]

### Step 2: Calculate the Distances Between Each Interval
Now we will calculate the distance traveled between each time interval:

1. **From \( t = 0 \) to \( t = 1 \):**
\[ \text{Distance} = |S(1) - S(0)| = |4 - 0| = 4 \text{ meters} \]

2. **From \( t = 1 \) to \( t = 2 \):**
\[ \text{Distance} = |S(2) - S(1)| = |2 - 4| = 2 \text{ meters} \]

3. **From \( t = 2 \) to \( t = 3 \):**
\[ \text{Distance} = |S(3) - S(2)| = |0 - 2| = 2 \text{ meters} \]

4. **From \( t = 3 \) to \( t = 4 \):**
\[ \text{Distance} = |S(4) - S(3)| = |4 - 0| = 4 \text{ meters} \]

5. **From \( t = 4 \) to \( t = 5 \):**
\[ \text{Distance} = |S(5) - S(4)| = |20 - 4| = 16 \text{ meters} \]

### Step 3: Total Distance Traveled
Now, we will sum up all the distances calculated:

\[ \text{Total Distance} = 4 + 2 + 2 + 4 + 16 = 28 \text{ meters} \]

### Final Answer
The total distance traveled by the particle during the first five seconds is **28 meters**.

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