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 of when the force on the boat becomes zero, we will follow these steps:

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

### Step 2: Differentiate the velocity function to find acceleration
Next, we differentiate the velocity function \( v(t) \) to find the acceleration \( a(t) \):
\[ a(t) = \frac{dv}{dt} = \frac{d}{dt}(2.4t - 0.6t^2) \]
Again using the power rule:
\[ a(t) = 2.4 - 2 \cdot 0.6 t = 2.4 - 1.2t \]

### Step 3: Calculate the force using Newton's second law
According to Newton's second law, force \( F \) is given by:
\[ F = m \cdot a \]
where \( m \) is the mass of the boat. Given that the mass \( m = 300 \) kg, we have:
\[ F(t) = 300 \cdot (2.4 - 1.2t) \]

### Step 4: Set the force equal to zero and solve for \( t \)
We want to find when the force becomes zero:
\[ 300 \cdot (2.4 - 1.2t) = 0 \]
Dividing both sides by 300:
\[ 2.4 - 1.2t = 0 \]
Now, solving for \( t \):
\[ 1.2t = 2.4 \\ t = \frac{2.4}{1.2} = 2 \text{ seconds} \]

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

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