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### Given Data:
- **Initial horizontal velocity of the airplane** (\( v_x \)) = 100 m/s.
- **Initial height** (\( h \)) = 2000 m.
- **Acceleration due to gravity** (\( g \)) = 9.8 m/s² (assuming no air resistance).

The problem consists of three parts:

---

### (a) Time required for the box to reach the earth
To find the time required for the box to reach the ground, we consider only the vertical motion, as horizontal and vertical motions are independent.

The initial vertical velocity (\( v_{y0} \)) is 0 (since it’s dropped without initial vertical speed).

Using the equation:
\[
h = \frac{1}{2} g t^2
\]
we can rearrange to solve for \( t \):
\[
t = \sqrt{\frac{2h}{g}}
\]

Substituting the values:
\[
t = \sqrt{\frac{2 \times 2000}{9.8}}
\]
\[
t \approx 20.2 \, \text{seconds}
\]

---

### (b) Horizontal distance traveled while falling
To find the horizontal distance traveled, we use the horizontal velocity and the time calculated above.

\[
\text{Horizontal distance} = v_x \times t
\]
Substituting the values:
\[
\text{Horizontal distance} = 100 \times 20.2
\]
\[
\text{Horizontal distance} \approx 2020 \, \text{meters}
\]

---

### (c) Horizontal and vertical components of the velocity when it strikes the earth
- **Horizontal component**: The horizontal velocity \( v_x \) remains constant, as there is no horizontal acceleration.
  \[
  v_x = 100 \, \text{m/s}
  \]

- **Vertical component**: The vertical velocity at the time of impact \( v_y \) can be calculated using:
  \[
  v_y = g \cdot t
  \]
  Substituting the values:
  \[
  v_y = 9.8 \times 20.2
  \]
  \[
  v_y \approx 198 \, \text{m/s}
  \]

---

### Summary of Answers:
1. **Time to reach the earth**: ~20.2 seconds.
2. **Horizontal distance traveled**: ~2020 meters.
3. **Horizontal component of velocity**: 100 m/s.
4. **Vertical component of velocity upon impact**: ~198 m/s.

Would you like further details on any step, or have additional questions?

---

### Follow-up Questions:
1. How would air resistance affect the time it takes for the box to reach the ground?
2. What would be the impact speed of the box, combining both horizontal and vertical components?
3. How would the answer change if the airplane were flying at a different altitude?
4. What if the airplane had an initial upward or downward vertical velocity?
5. How would the problem change if we considered a varying gravitational field?

### Tip:
When solving projectile motion problems, always separate horizontal and vertical motions—they are independent of each other!

An airplane flying at 100 m/s drops a box at an elevation of 2000 m. (a) How much time is required for the box to reach the earth? (b) How far does it travel horizontally while falling? (c) Find the horizontal and vertical components of its velocity when it strikes the earth.

Solve a projectile motion problem involving an airplane flying at 100 m/s dropping a box from an elevation of 2000 m. This step-by-step solution covers calculating the time for the box to reach the ground, the horizontal distance traveled, and the velocity components at impact.

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