The distance travelled by a particle in time t is given by `x = kt^3` , where `k = 10 m s^(-3)` . The average speed of the particle from t = 1 s to t = 5 s is 

Options

A. `280 ms^(-1)`
B. `300 ms^(-1)`
C. `310 ms^(-1)`
D. `320 ms^(-1)`

Answer

To find the average speed of the particle from \( t = 1 \, \text{s} \) to \( t = 5 \, \text{s} \), we will follow these steps:

### Step 1: Understand the given equation
The distance traveled by the particle is given by the equation:
\[ x = kt^3 \]
where \( k = 10 \, \text{m/s}^3 \).

### Step 2: Calculate the distance at \( t = 1 \, \text{s} \)
Substituting \( t = 1 \) into the equation:
\[ x_1 = k(1)^3 = 10 \times 1^3 = 10 \, \text{m} \]

### Step 3: Calculate the distance at \( t = 5 \, \text{s} \)
Now substituting \( t = 5 \) into the equation:
\[ x_2 = k(5)^3 = 10 \times 5^3 = 10 \times 125 = 1250 \, \text{m} \]

### Step 4: Calculate the change in distance
The change in distance (\( \Delta x \)) is:
\[ \Delta x = x_2 - x_1 = 1250 \, \text{m} - 10 \, \text{m} = 1240 \, \text{m} \]

### Step 5: Calculate the time interval
The time interval (\( \Delta t \)) is:
\[ \Delta t = 5 \, \text{s} - 1 \, \text{s} = 4 \, \text{s} \]

### Step 6: Calculate the average speed
The average speed (\( V_{\text{avg}} \)) is given by the formula:
\[ V_{\text{avg}} = \frac{\Delta x}{\Delta t} = \frac{1240 \, \text{m}}{4 \, \text{s}} = 310 \, \text{m/s} \]

### Final Answer
The average speed of the particle from \( t = 1 \, \text{s} \) to \( t = 5 \, \text{s} \) is:
\[ \boxed{310 \, \text{m/s}} \]

---