Yes! Let's break this down step by step.

Given Information:

• The airliner is flying at a constant speed of 850 km/h.
• The rocket is dropped from the airliner at 12.0 km altitude.
• After a brief free fall, the rocket's motor turns on.
• Once the motor starts, the rocket accelerates at 3.00g ($3 \times 9.81$ m/s²) at an angle of 30.0° above the horizontal.
• For safety, the rocket should be at least 1.00 km in front of the airliner when it climbs through the airliner's altitude.
• Air resistance is ignored.

Step 1: Convert Given Values to SI Units

• Speed of airliner:
  $850 \text{ km/h} = \frac{850 \times 1000}{3600} = 236.11 \text{ m/s}$
• Distance for safety:
  $1.00 \text{ km} = 1000 \text{ m}$
• Acceleration of rocket: $3.00 \times 9.81 = 29.43 \text{ m/s}^2$

Step 2: Break Acceleration into Components

• The rocket's acceleration has both horizontal and vertical components: $a_x = 29.43 \cos(30^\circ) = 25.48 \text{ m/s}^2$ $a_y = 29.43 \sin(30^\circ) = 14.72 \text{ m/s}^2$

Step 3: Find Time Required to Move 1 km Ahead

The horizontal displacement $x$ of the rocket is given by: $x = v_0 t + \frac{1}{2} a_x t^2$ Since the rocket is initially dropped from the plane, its initial horizontal velocity is the same as the airliner: $v_0 = 236.11 \text{ m/s}$ Setting $x = 1000$ m: $1000 = 236.11 t + \frac{1}{2} (25.48) t^2$ $1000 = 236.11 t + 12.74 t^2$ Rearrange into quadratic form: $12.74 t^2 + 236.11 t - 1000 = 0$

Step 4: Solve for $t$

Using the quadratic formula: $t = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ where

• $a = 12.74$
• $b = 236.11$
• $c = -1000$

$t = \frac{-236.11 \pm \sqrt{(236.11)^2 - 4(12.74)(-1000)}}{2(12.74)}$

$t = \frac{-236.11 \pm \sqrt{55760.69 + 50960}}{25.48}$

$t = \frac{-236.11 \pm \sqrt{106720.69}}{25.48}$

$t = \frac{-236.11 \pm 326.7}{25.48}$

Taking the positive root:

$t = \frac{-236.11 + 326.7}{25.48}$

$t = \frac{90.59}{25.48} = 3.56 \text{ s}$

Answer:

The minimum time the rocket must fall before its engine starts is 3.56 seconds.

Let me know if you need further details or explanation!

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Related Questions:

1. How would air resistance affect the rocket’s motion?
2. What would happen if the acceleration angle were greater than 30°?
3. How does the rocket’s vertical motion change the time calculation?
4. What if the airliner’s speed were different? How would that change the result?
5. Can we derive a general formula for this type of problem?

Tip:

Always convert units to SI (meters, seconds) before solving physics problems! 🚀