Question

2. Napoleon Crossing the Alps Napoleon's route through the Alps involved travelling through the Grand St Bernard Pass, a climb of 6km and a descent of 4km. His army travelled twice as fast downhill as it did uphill and the whole journey took 8 hours. How long did it take him to reach the top?​

Answer 1

The speed of the army up the hill and the distance travelled by the army

up the hill determines the time the army takes to reach the top.

Response:

• The time it takes the army to reach the to of the hill is 6 hours

Which method is used for the finding the time to travel up the hill?

Given:

Distance of the climb = 6 km

Distance of the descent = 4 km

Speed of the army downhill = 2 × The speed uphill

Duration of the journey = 8 hours

Required:

The time it takes the army to reach the top.

Solution:

Let v represent the speed of the army on the way up, we have;

The speed downhill = 2·v

Distance, d= Speed, v × Time, t

$t = \mathbf{\dfrac{d}{v}}$

Which gives;

$Time \ to \ the \ top, \ t_1 =\mathbf{\dfrac{6}{v}}$

$Time \ downhill, \ t_2 = \mathbf{\dfrac{4}{2 \cdot v}}$

• t₁ + t₂ = 8

Which gives;

$\mathbf{\dfrac{6}{v} + \dfrac{4}{2 \cdot v}} = 8$

$\dfrac{6 \times 2 + 4}{2 \cdot v} =\dfrac{16}{2 \cdot v} = 8$

2·v × 8 = 16

$v = \dfrac{16}{16} = 1$

The speed of the army uphill, v = 1 km/hr

Therefore;

• $Time \ to \ the \ top, \ t_1 =\dfrac{6 \, km}{1 \, km/hr} = \underline{6 \ hour}$

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