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To solve this problem, we need to find the components of the net force on the airplane from the given momentum equation.

### Given:
Momentum \( \vec{p}(t) \) is provided as:
\[
\vec{p}(t) = \left( -0.75 \, \text{kg} \cdot \text{m/s}^3 \right)t^2 \hat{i} + \left( 3.0 \, \text{kg} \cdot \text{m/s} \right) \hat{i} + \left( 0.25 \, \text{kg} \cdot \text{m/s}^2 \right)t \hat{j}
\]

### Unknown:
The net force \( \vec{F}(t) \), which has x-, y-, and z-components (if applicable).

### Equation:
Force is related to momentum by the equation:
\[
\vec{F}(t) = \frac{d\vec{p}(t)}{dt}
\]
Thus, we can find the net force by differentiating the momentum equation with respect to time.

### Solution:
Let's differentiate each term of the momentum function:

1. **For the \( \hat{i} \) (x-component) term:**
   \[
   \frac{d}{dt}\left( -0.75 \, \text{kg} \cdot \text{m/s}^3 \cdot t^2 + 3.0 \, \text{kg} \cdot \text{m/s} \right) 
   = -0.75 \cdot 2t \, \text{kg} \cdot \text{m/s}^3 = -1.5t \, \text{N}
   \]
   The constant term \( 3.0 \, \text{kg} \cdot \text{m/s} \) disappears since the derivative of a constant is zero.

2. **For the \( \hat{j} \) (y-component) term:**
   \[
   \frac{d}{dt}\left( 0.25 \, \text{kg} \cdot \text{m/s}^2 \cdot t \right) = 0.25 \, \text{kg} \cdot \text{m/s}^2 = 0.25 \, \text{N}
   \]

### Final Answer:
The net force \( \vec{F}(t) \) is:
\[
\vec{F}(t) = (-1.5t) \hat{i} + (0.25) \hat{j}
\]

Thus, the x-component of the net force is \( -1.5t \, \text{N} \), the y-component is \( 0.25 \, \text{N} \), and there is no z-component (since the momentum expression doesn't include any \( \hat{k} \)-terms).

Do you have any questions, or would you like more details on the differentiation process?

### Related Questions:
1. How does force relate to the rate of change of momentum in general?
2. What would happen if the momentum had a \( z \)-component in this problem?
3. How can you determine the acceleration of the airplane using the force found?
4. What physical meaning does the constant \( 3.0 \, \text{kg} \cdot \text{m/s} \) term have in the momentum equation?
5. How would you interpret the units of the various terms in the momentum equation?

**Tip:** Always make sure the units of momentum are consistent (e.g., kg·m/s) before performing any operations on them.

A radio-controlled model airplane has a momentum given by [(−0.75 kg·m/s³)t² + (3.0 kg·m/s)]î + (0.25 kg·m/s²)tĵ. What are the x-, y-, and z-components of the net force on the airplane?

This problem covers how to calculate the x- and y-components of the net force from a given momentum equation using differentiation. Learn to apply Newton's Second Law in vector form to find force components for a radio-controlled model airplane.

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