Question

Consider a train which can accelerate with an acceleration of \[20\,{\text{cm/}}{{\text{s}}^{\text{2}}}\] and slow down with deceleration of \[100\,{\text{cm/}}{{\text{s}}^{\text{2}}}\]. Find the minimum time for the train to travel between the stations \[2.7\,{\text{km}}\] apart.
A. \[90\,{\text{s}}\]
B. \[180\,{\text{s}}\]
C. \[{\text{160}}\,{\text{s}}\]
D. \[{\text{240}}\,{\text{s}}\]

Answer 1

Hint: We use the formula of maximum speed, deceleration, displacement and use the given information for this solution and solve it.

Complete step by step answer:
For maximum speed,
 $ v = u + at \\ = 0.2\,t \\
v = 0.2\,{t_1} $ …… (1)
For deceleration,
  $ v = u - at \\
  0 = v - 1.0\,{t_2} \\
v = {t_2} $ …… (2)
So,
  $ {t_1} = \dfrac{{{t_2}}}{{0.2}} \\
   = 5\,{t_2} \\ $
And,
For displacement,
$ S = ut + \dfrac{1}{2}a{t^2} \\
  S = 2700\,{\text{m}} \\ $
Then,
\[2700 = 0.1\left[ {{{\left( {5{t_2}} \right)}^2} + {{\left( {{t_2}} \right)}^2} - \dfrac{1}{2}{t_2}^2} \right]\]
Here,
$ r = 80\,{\text{cm}}$
\[{\text{3}}{{\text{t}}_2}^2 = 2700\]
\[{t_2} = 30\,\sec \]
\[{t_1} = 150\,\sec \]
So, the total time is,
 ${t_1} + {t_2} = 150 + 30 \\ = 180\,\sec \\$
Hence, the required answer is \[180\,\sec \].

So, the correct answer is “Option B”.

Additional Information:
Acceleration: Acceleration is the rate at which an object's speed in relation to time varies. Accelerations (magnitude and direction) are vector quantities. The direction of the acceleration of an object is given by the direction of the net force acting on it. As defined in Newton's Second Law, the magnitude of the object's acceleration is the combined effect of two reasons:
- The net balance of all external forces acting on that object - the magnitude of this resultant force is directly proportional.
- The weight of this object is inversely proportional to the mass of the object, depending on the materials it is made of.

Deceleration: Acceleration is strictly defined as the speed vector change rate. On the other hand, deceleration is acceleration which causes a "speed" reduction. The acceleration deceleration isn't the reverse. It's definitely not a negative rate of speed change.

Displacement: A displacement is the vector with the shortest length between the original and the final positions of the motion of a point P. It can be found in a straight line from the initial location to the final position of the point’s path both the distance and the direction of the net or complete motion. The translation that maps the initial position to the final position will define a displacement.

Note:
We found that an object's acceleration depends on the object's mass and the force's scale. Say that an object is accelerated by increasing force and decreases by increasing mass.