A particle moves along a straight line on y-axis. The distance of the particle from O varies with time and is given by : `y = 20t - t^2`. The distance travelled by the particle before it momentarily comes to rest is

Options

A. 200 m
B. 100 m
C. 50 m
D. 150 m

Answer

To solve the problem, we need to find the distance traveled by the particle before it comes to rest. The distance as a function of time is given by the equation:

\[ y = 20t - t^2 \]

### Step 1: Find the velocity of the particle
To find when the particle comes to rest, we first need to determine its velocity. The velocity \( v \) is the derivative of the distance \( y \) with respect to time \( t \).

\[ v = \frac{dy}{dt} = \frac{d}{dt}(20t - t^2) \]

### Step 2: Differentiate the equation
Now, we differentiate the equation:

\[ v = 20 - 2t \]

### Step 3: Set the velocity to zero
To find the time when the particle comes to rest, we set the velocity \( v \) to zero:

\[ 0 = 20 - 2t \]

### Step 4: Solve for time \( t \)
Rearranging the equation gives us:

\[ 2t = 20 \]
\[ t = \frac{20}{2} = 10 \text{ seconds} \]

### Step 5: Calculate the distance traveled at \( t = 10 \) seconds
Now that we know the time at which the particle comes to rest, we can substitute \( t = 10 \) seconds back into the original distance equation to find the distance traveled:

\[ y = 20(10) - (10)^2 \]
\[ y = 200 - 100 \]
\[ y = 100 \text{ meters} \]

### Final Answer
The distance traveled by the particle before it momentarily comes to rest is **100 meters**.

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