The potential energy of a 4 kg particle free to move along the x-axis is given as `U(x) = (x^(3))/(3)- (5x^(2))/(2)+6x+3` Total mechanical energy of the particle is 17J. Then find the maximum kinetic energy of the particle

Explanation

To find the maximum kinetic energy of the particle given its potential energy function and total mechanical energy, we can follow these steps:

### Step 1: Understand the relationship between total mechanical energy, kinetic energy, and potential energy.
The total mechanical energy (E) of a system is the sum of its kinetic energy (K) and potential energy (U):
\[ E = K + U \]
From this, we can express the kinetic energy as:
\[ K = E - U \]

### Step 2: Identify the given values.
From the problem, we know:
- Total mechanical energy, \( E = 17 \, \text{J} \)
- Potential energy function, \( U(x) = \frac{x^3}{3} - \frac{5x^2}{2} + 6x + 3 \)

### Step 3: Find the critical points of the potential energy function.
To find the maximum kinetic energy, we need to determine where the potential energy is at a minimum. We do this by finding the derivative of the potential energy function and setting it to zero:
\[ \frac{dU}{dx} = 0 \]

Calculating the derivative:
\[ \frac{dU}{dx} = x^2 - 5x + 6 \]

Setting the derivative to zero:
\[ x^2 - 5x + 6 = 0 \]

### Step 4: Solve the quadratic equation.
Factoring the equation:
\[ (x - 2)(x - 3) = 0 \]
Thus, the critical points are:
\[ x = 2 \quad \text{and} \quad x = 3 \]

### Step 5: Determine whether these points are minima or maxima.
To find out if these critical points correspond to a minimum or maximum, we calculate the second derivative:
\[ \frac{d^2U}{dx^2} = 2x - 5 \]

Now we evaluate the second derivative at the critical points:
1. For \( x = 2 \):
\[ \frac{d^2U}{dx^2} = 2(2) - 5 = -1 \quad (\text{maximum}) \]
2. For \( x = 3 \):
\[ \frac{d^2U}{dx^2} = 2(3) - 5 = 1 \quad (\text{minimum}) \]

### Step 6: Calculate the minimum potential energy.
Now we need to find the potential energy at the minimum point \( x = 3 \):
\[ U(3) = \frac{3^3}{3} - \frac{5(3^2)}{2} + 6(3) + 3 \]
Calculating this:
\[ U(3) = \frac{27}{3} - \frac{45}{2} + 18 + 3 \]
\[ U(3) = 9 - 22.5 + 18 + 3 \]
\[ U(3) = 7.5 \, \text{J} \]

### Step 7: Calculate the maximum kinetic energy.
Using the total mechanical energy:
\[ K = E - U \]
Substituting the values:
\[ K = 17 \, \text{J} - 7.5 \, \text{J} = 9.5 \, \text{J} \]

### Final Answer:
The maximum kinetic energy of the particle is:
\[ \boxed{9.5 \, \text{J}} \]