To calculate the time it takes for the vehicle to travel $403.3 \, \text{m}$, we need to use the given data and apply the kinematic equations. Let's break this problem into steps.

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Given Data:

1. Mass of the vehicle, $m = 500 \, \text{kg}$
2. Force applied, $F = 800,000 \, \text{N}$ (converted from $800 \, \text{kN}$)
3. Distance traveled, $s = 403.3 \, \text{m}$
4. Initial velocity, $u = 0 \, \text{m/s}$

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Step 1: Calculate Acceleration ($a$)

Using Newton's second law: $F = m \cdot a$ $a = \frac{F}{m} = \frac{800,000}{500} = 1600 \, \text{m/s}^2$

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Step 2: Use the Kinematic Equation to Find Time ($t$)

The kinematic equation is: $s = ut + \frac{1}{2} a t^2$ Since $u = 0$, this simplifies to: $s = \frac{1}{2} a t^2$ Substitute $s = 403.3 \, \text{m}$ and $a = 1600 \, \text{m/s}^2$: $403.3 = \frac{1}{2} \cdot 1600 \cdot t^2$ $403.3 = 800 t^2$ $t^2 = \frac{403.3}{800}$ $t^2 = 0.504125$ $t = \sqrt{0.504125} \approx 0.71 \, \text{s}$

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Final Answer:

The time it takes for the vehicle to travel $403.3 \, \text{m}$ is approximately: $t \approx 0.71 \, \text{seconds}$

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Do you want a detailed explanation or further clarification?
Here are 5 follow-up questions related to this calculation:

1. What would the time be if the force applied was doubled?
2. How would the time change if the mass of the vehicle was increased to 1000 kg?
3. What is the final velocity of the vehicle after traveling 403.3 m?
4. How can this scenario be applied to real-world engineering problems?
5. What assumptions are made in this calculation, and how could they affect the result?

Tip: Always check units and convert them to SI before performing calculations for consistency.