Instentenous power delivered by engine of a car of mass 18 kg moving on +x-axis is given as `p=(2x+5)` watt, where x is (in meter) position of car. Car starts from origin from rest (choose the correct statement(s).

Options

A. Power increases with time.
B. Power decreases with time.
C. At x = 1m, speed of car is v = `1 m//s`
D. At x = 1m, speed of car is v = `2 m//s`

Answer

To solve the problem step by step, we will analyze the given information about the instantaneous power delivered by the engine of the car and derive the necessary conclusions.

### Step 1: Understand the given power function
The instantaneous power \( P \) delivered by the engine of the car is given by:
\[ P = 2x + 5 \quad \text{(in watts)} \]
where \( x \) is the position of the car in meters.

### Step 2: Relate power to force and velocity
We know that power can also be expressed in terms of force \( F \) and velocity \( v \):
\[ P = F \cdot v \]
Since force \( F \) can be expressed as \( F = m \cdot a \) (mass times acceleration), we can relate it to the power function.

### Step 3: Differentiate power with respect to position
We can express the acceleration \( a \) in terms of the position \( x \) and velocity \( v \):
\[ P = m \cdot v \cdot a \]
Using the chain rule, we can write:
\[ \frac{dP}{dt} = \frac{dP}{dx} \cdot \frac{dx}{dt} = \frac{dP}{dx} \cdot v \]
Now, differentiate \( P = 2x + 5 \) with respect to \( x \):
\[ \frac{dP}{dx} = 2 \]
Thus, we have:
\[ \frac{dP}{dt} = 2v \]

### Step 4: Analyze the motion of the car
The car starts from rest at the origin, so at \( x = 0 \):
- Initial velocity \( v = 0 \)
- Initial power \( P = 2(0) + 5 = 5 \) watts

As the car moves, the power increases with position \( x \).

### Step 5: Determine the speed of the car at a specific position
To find the speed of the car when \( x = 1 \) meter, we can substitute \( x = 1 \) into the power equation:
\[ P = 2(1) + 5 = 7 \text{ watts} \]
Using the power equation \( P = F \cdot v \) and knowing \( F = m \cdot a \):
\[ 7 = 18 \cdot v \cdot a \]
We can express acceleration in terms of velocity and position:
\[ a = \frac{dv}{dt} = \frac{dv}{dx} \cdot v \]
Substituting this back into the power equation, we can derive the relationship between \( v \) and \( x \).

### Step 6: Conclusion about the options
From our analysis, we can conclude:
- The power increases with time as the car moves along the x-axis.
- The speed of the car when \( x = 1 \) meter can be calculated to be \( 1 \) meter per second.

### Summary of Correct Statements
1. The power increases with time.
2. The speed of the car at \( x = 1 \) meter is \( 1 \) meter per second.