A particle moves along a straight line AB. At a time t (in seconds) the distance x (in metres) of the particle from O is given by x = 600 + 12t – `t^3`. How long would the particle travel before coming to rest: -

Options

A. 24m
B. 40m
C. 56m
D. 16m

Explanation

To solve the problem step by step, we need to find out how long the particle travels before coming to rest, and then calculate the distance it travels during that time.

### Step 1: Determine the velocity function
The position of the particle as a function of time \( t \) is given by:

\[ x(t) = 600 + 12t - t^3 \]

To find the velocity \( v(t) \), we take the derivative of the position function with respect to time \( t \):

\[ v(t) = \frac{dx}{dt} = \frac{d}{dt}(600 + 12t - t^3) \]

Calculating the derivative:

\[ v(t) = 0 + 12 - 3t^2 = 12 - 3t^2 \]

### Step 2: Set the velocity to zero to find when the particle comes to rest
The particle comes to rest when the velocity is zero:

\[ 12 - 3t^2 = 0 \]

Solving for \( t \):

\[ 3t^2 = 12 \\ t^2 = \frac{12}{3} = 4 \\ t = 2 \text{ seconds} \]

### Step 3: Calculate the distance traveled before coming to rest
Now that we know the time when the particle comes to rest is \( t = 2 \) seconds, we need to find the distance traveled during this time.

We will calculate the position at \( t = 2 \) seconds:

\[ x(2) = 600 + 12(2) - (2)^3 \]

Calculating \( x(2) \):

\[ x(2) = 600 + 24 - 8 = 616 \text{ meters} \]

### Step 4: Calculate the initial position at \( t = 0 \)
Now we will find the position at \( t = 0 \):

\[ x(0) = 600 + 12(0) - (0)^3 = 600 \text{ meters} \]

### Step 5: Calculate the distance traveled
The distance traveled by the particle before coming to rest is:

\[ \text{Distance} = x(2) - x(0) = 616 - 600 = 16 \text{ meters} \]

Thus, the particle travels a distance of **16 meters** before coming to rest.

### Summary of Steps:
1. Find the velocity function by differentiating the position function.
2. Set the velocity to zero to find the time when the particle comes to rest.
3. Calculate the position at the time of rest.
4. Calculate the initial position.
5. Subtract the initial position from the position at rest to find the distance traveled.