Answer to Question #145213 in Classical Mechanics for Isurini Sirisena

Question

There has been a proposal to build a train tunnel
underneath the Atlantic Ocean from England to America.
The suggestion is that in the future the trip of 5000 km
could take as little as one hour.
Assume that half the time is spent accelerating uniformly
and the other half is spent decelerating uniformly with the
same magnitude as the acceleration.
(a) Show that the acceleration would be about 2 m s−2. [2]
(b) Calculate the maximum speed. [2]
(c) Calculate the resultant force required to decelerate
the train.
mass of train = 4.5 × 105 kg [2]

Answer 1

Explanations & Calculations

• By considering the symmetry of the motion the distance the train travels during the period of acceleration & deceleration should be equal & therefore is 2500km
• Applying $\small s=ut+\frac{1}{2}at^2$ for the period of acceleration,

$\qquad\qquad \begin{aligned} \small 2500km&= \small 0t+0.5\times a\times (0.5h)^2\\ \small a &= \small 20000kmh^{-2}\\ \small a &= \small 20000\times (10^3)m\times (3600s)^{-2}\\ \small a &= \small 1.54 m^{-2}\approx2ms^{-2 } \end{aligned}$

• Applying $\small v=u+at$ for the period of acceleration

$\qquad\qquad \begin{aligned} \small V_{max}&= \small 0+20000kmh^{-2 }\times 0.5h\\ &= \small \bold{10^4kmh^{-1}}\approx 2.8kms^{-1} \end{aligned}$

• Applying Newton's second law towards the moving direction during the period of deceleration,

$\qquad\qquad \begin{aligned} \small |F|=m|a|\\ &= \small 4.5\times 10^5kg\times 1.54ms^-2\\ &= \small \bold{693kN} \end{aligned}$