A boat of mass 300 kg moves according to the equation `x= 1.2t^(2) - 0.2 t^(3)`. When the force will become zero ?

Options

A. 2s
B. 1s
C. 6s
D. 2.8 s

Answer

To solve the problem, we need to determine when the force acting on the boat becomes zero. According to Newton's second law, the force \( F \) is given by:

\[ F = m \cdot \frac{dv}{dt} \]

where \( m \) is the mass of the boat, and \( \frac{dv}{dt} \) is the acceleration (the rate of change of velocity).

### Step 1: Differentiate the position function to find velocity
The position of the boat is given by the equation:

\[ x(t) = 1.2t^2 - 0.2t^3 \]

To find the velocity \( v(t) \), we differentiate the position function with respect to time \( t \):

\[ v(t) = \frac{dx}{dt} = \frac{d}{dt}(1.2t^2 - 0.2t^3) \]

Using the power rule of differentiation:

\[ v(t) = 2 \cdot 1.2t^{2-1} - 3 \cdot 0.2t^{3-1} = 2.4t - 0.6t^2 \]

### Step 2: Differentiate the velocity function to find acceleration
Next, we differentiate the velocity function to find the acceleration \( a(t) \):

\[ a(t) = \frac{dv}{dt} = \frac{d}{dt}(2.4t - 0.6t^2) \]

Again, using the power rule:

\[ a(t) = 2.4 - 1.2t \]

### Step 3: Set acceleration equal to zero
To find when the force becomes zero, we need to set the acceleration equal to zero:

\[ a(t) = 0 \implies 2.4 - 1.2t = 0 \]

### Step 4: Solve for time \( t \)
Now, we solve for \( t \):

\[ 1.2t = 2.4 \implies t = \frac{2.4}{1.2} = 2 \text{ seconds} \]

### Conclusion
The force acting on the boat will become zero at \( t = 2 \) seconds.