If acceleration gradient of a moving body along positive x-axis is `-4s^(-2)` and body starts its motion from rest from `x = 5m`, having maximum acceleration (magnitude) then velocity of moving body at `x = -3m` is

Options

A. `6ms^(-1)`
B. `4 ms^(-1)`
C. `8 ms^(-1)`
D. Can't reach at `x = -3m`

Explanation

To find the velocity of the moving body at \( x = -3 \, m \), we can follow these steps:

### Step 1: Understand the given information
We are given that the acceleration gradient \( A \) is \( -4x^{-2} \). This means the acceleration \( a \) can be expressed as:
\[ a = -4x \]
The body starts from rest at \( x = 5 \, m \), which means the initial velocity \( v_0 = 0 \).

### Step 2: Use the relationship between acceleration, velocity, and position
We can use the formula that relates acceleration, velocity, and position:
\[ a = v \frac{dv}{dx} \]
Substituting the expression for acceleration:
\[ -4x = v \frac{dv}{dx} \]

### Step 3: Rearrange and integrate
Rearranging gives:
\[ v \, dv = -4x \, dx \]
Now we integrate both sides. The left side integrates with respect to \( v \) and the right side with respect to \( x \):
\[ \int v \, dv = \int -4x \, dx \]

### Step 4: Perform the integration
The left side integrates to:
\[ \frac{v^2}{2} \]
The right side integrates to:
\[ -2x^2 + C \]
Thus, we have:
\[ \frac{v^2}{2} = -2x^2 + C \]

### Step 5: Determine the constant of integration \( C \)
To find \( C \), we use the initial condition. At \( x = 5 \, m \), \( v = 0 \):
\[ \frac{0^2}{2} = -2(5^2) + C \]
\[ 0 = -50 + C \implies C = 50 \]

### Step 6: Substitute \( C \) back into the equation
Now we substitute \( C \) back into our equation:
\[ \frac{v^2}{2} = -2x^2 + 50 \]

### Step 7: Solve for \( v^2 \)
Multiplying through by 2 gives:
\[ v^2 = -4x^2 + 100 \]

### Step 8: Find the velocity at \( x = -3 \, m \)
Now we substitute \( x = -3 \, m \):
\[ v^2 = -4(-3)^2 + 100 \]
\[ v^2 = -4(9) + 100 \]
\[ v^2 = -36 + 100 = 64 \]

### Step 9: Take the square root to find \( v \)
Taking the square root gives:
\[ v = \sqrt{64} = 8 \, m/s \]

### Final Answer
The velocity of the moving body at \( x = -3 \, m \) is \( 8 \, m/s \).

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