Relative Motion: Car vs Truck vs Bridge Lengths

We are told:

• A car takes 10 seconds to overtake a truck.
• The same car takes 20 seconds to cross a bridge.

We are asked:

Which is longer — the truck or the bridge?

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1. Modeling the situation

Assume the car moves with constant speed relative to the ground. Let:

• Car speed: $v$
• Truck length: $L_t$
• Bridge length: $L_b$

When the car overtakes the truck (from behind):

• For a complete overtake, the car must move its entire length relative to the truck's rear plus the truck's full length.
• If we ignore the car's own length (or assume it’s the same in both comparisons so it cancels), the essential distance that encodes the comparison is the other object’s length.

In many standard textbook formulations, especially for which is longer comparisons, the key idea is:

Time to pass an object at constant speed is proportional to the object's length.

So:

• Time to pass truck: $t_t = 10\,\text{s}$
• Time to pass bridge: $t_b = 20\,\text{s}$

If the car's speed $v$ is the same in both cases, then:

$$L_t \propto t_t, \quad L_b \propto t_b$$

Thus:

$$\frac{L_b}{L_t} = \frac{t_b}{t_t} = \frac{20}{10} = 2$$

So the bridge is twice as long as the truck.

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2. Answer

• The bridge is longer than the truck (specifically, about twice as long in this scenario).

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3. What the visualization shows

This interactive visualization lets you:

• Adjust the car's speed.
• Adjust the truck and bridge crossing times.
• See how the implied lengths of the truck and bridge compare.

On the canvas:

• A horizontal track centered in the screen represents the road.
• A cyan bar (car) moves along the track.
• A pink segment marks the truck region.
• A yellow segment marks the bridge region.
• Above, simple proportional bars show the relative lengths computed from $L \propto v \times t$.

We normalize everything so:

$$L_t = v \cdot t_t, \quad L_b = v \cdot t_b$$

Only the ratio $L_b / L_t = t_b / t_t$ matters for which is longer.

You can change times and speed and visually see:

• If the bridge bar is longer, the bridge is longer.
• If the truck bar is longer, the truck is longer.

The default values recreate the problem statement: 10 s for the truck and 20 s for the bridge, clearly showing the bridge as longer.