Question 25: Motion in 1D

Question

Two trains 'A' and 'B' of length '$$l$$' and '$$4 l$$' are travelling into a tunnel of length '$$\mathrm{L}$$' in parallel tracks from opposite directions with velocities $$108 \mathrm{~km} / \mathrm{h}$$ and $$72 \mathrm{~km} / \mathrm{h}$$, respectively. If train 'A' takes $$35 \mathrm{~s}$$ less time than train 'B' to cross the tunnel then. length '$$L$$' of tunnel is :
(Given $$\mathrm{L}=60 l$$ )

Accepted Answer

Let's start by converting the velocities of both trains to m/s:
Train A: $$108 \frac{km}{h} 	imes \frac{1000 m}{km} 	imes \frac{1 h}{3600 s} = 30 \frac{m}{s}$$
Train B: $$72 \frac{km}{h} 	imes \frac{1000 m}{km} 	imes \frac{1 h}{3600 s} = 20 \frac{m}{s}$$
To cross the tunnel, Train A has to cover a distance equal to the length of the tunnel plus its own length: $$L + l$$
Similarly, Train B has to cover a distance equal to the length of the tunnel plus its own length: $$L + 4l$$
We are given that Train A takes 35 seconds less time than Train B to cross the tunnel. Let's denote the time taken by Train A as $$t_A$$ and the time taken by Train B as $$t_B$$. Then, we have:
$$t_B = t_A + 35$$
Using the formula distance = velocity ÃÂ time, we can write the equations for both trains:
Train A: $$(L + l) = 30t_A$$
Train B: $$(L + 4l) = 20t_B$$
Now, we can substitute $$t_B = t_A + 35$$ in the equation for Train B:
$$(L + 4l) = 20(t_A + 35)$$
We have two equations and two unknowns ($$L$$ and $$t_A$$). We can solve this system of equations by eliminating one of the unknowns. Let's eliminate $$t_A$$ by expressing it from the equation for Train A:
$$t_A = \frac{L + l}{30}$$
Now, substitute this expression for $$t_A$$ in the equation for Train B:
$$(L + 4l) = 20\left(\frac{L + l}{30} + 35ight)$$
Multiplying both sides by 30 to get rid of the fraction:
$$30(L + 4l) = 20(L + l) + 20 	imes 35 	imes 30$$
Expanding the equation:
$$30L + 120l = 20L + 20l + 21,000$$
Simplify:
$$10L + 100l = 21,000$$
Since we are given that $$L = 60l$$, substitute this into the equation:
$$10(60l) + 100l = 21,000$$
Solve for $$l$$:
$$700l = 21,000$$
$$l = 30$$
Now, substitute the value of $$l$$ back into the equation for $$L$$:
$$L = 60l = 60 	imes 30 = 1800$$
So, the length of the tunnel is:
1800 m

Answer

900 m

Answer

1200 m

Answer

1800 m

Answer

2700 m

Motion in 1D

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