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 step by step, we will follow these instructions:

### Step 1: Understand the given equation
The distance of the particle from point O is given by the equation:
\[ y = 20t - t^2 \]
where \( y \) is the distance from O and \( t \) is the time in seconds.

**Hint:** Identify the variables and the relationship between distance and time.

### Step 2: Find the velocity of the particle
To find the velocity, we need to differentiate the distance equation with respect to time \( t \):
\[ v = \frac{dy}{dt} \]
Differentiating \( y = 20t - t^2 \):
\[ v = \frac{d}{dt}(20t - t^2) = 20 - 2t \]

**Hint:** Remember that velocity is the derivative of distance with respect to time.

### Step 3: Set the velocity to zero to find when the particle comes to rest
To find when the particle comes to rest, we set the velocity equation to zero:
\[ 20 - 2t = 0 \]
Solving for \( t \):
\[ 2t = 20 \]
\[ t = 10 \, \text{seconds} \]

**Hint:** The particle comes to rest when the velocity is zero.

### Step 4: Calculate the distance traveled at \( t = 10 \) seconds
Now, we need to find the position of the particle at \( t = 10 \) seconds using the original distance equation:
\[ y = 20(10) - (10)^2 \]
\[ y = 200 - 100 \]
\[ y = 100 \, \text{meters} \]

**Hint:** Substitute the value of \( t \) back into the original distance equation to find the position.

### Step 5: Calculate the initial position at \( t = 0 \)
Next, we calculate the position of the particle at \( t = 0 \):
\[ y = 20(0) - (0)^2 \]
\[ y = 0 \, \text{meters} \]

**Hint:** Always check the initial position to find the total distance traveled.

### Step 6: Determine the total distance traveled
The total distance traveled by the particle before it comes to rest is:
\[ \text{Distance} = y_{\text{final}} - y_{\text{initial}} \]
\[ \text{Distance} = 100 - 0 = 100 \, \text{meters} \]

**Hint:** The total distance is the difference between the final and initial positions.

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

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