Question

The acceleration of a train travelling with speed of 400 m/s as it goes round a curve of radius 160m, is
A. \[1{\rm{ km/}}{{\rm{s}}^2}\]
B. \[100{\rm{ m/}}{{\rm{s}}^2}\]
C. \[10{\rm{ m/}}{{\rm{s}}^2}\]
D. \[1{\rm{ m/}}{{\rm{s}}^2}\]

Answer 1

Hint:The centripetal acceleration is always directed radially towards the center of the circle with a magnitude which is equal to the square of the speed or velocity of the body along the curve divided by the total distance from the center of the circle to the moving body.

Formula used:
When an object is moving in a circular motion then its acceleration can be measured by using the following equation-
\[a = \dfrac{{{v^2}}}{r}\]
Where \[a\] is the centripetal acceleration, v is the velocity and r is the radius.

Complete step by step solution:
Given Speed of a train, \[v = 400{\rm{ m/s}}\]
Radius, \[r = 160{\rm{ m}}\]
As we know that centripetal acceleration is,
\[a = \dfrac{{{v^2}}}{r}\]
After substituting the values, we get
\[a = \dfrac{{{{(400)}^2}}}{{160}} \\
\Rightarrow a= {10^3}{\rm{ m/}}{{\rm{s}}^2} \\
\therefore a= 1\,{\rm{ km/}}{{\rm{s}}^2}\]
Therefore, the acceleration of a train travelling will be \[1{\rm{ km/}}{{\rm{s}}^2}\].

Hence option A is the correct answer.

Note: The centripetal acceleration is defined as the acceleration of a body travelling in a circular path. As velocity is a vector quantity it has both magnitude and direction. When a body travels on a circular path then its direction constantly changes and hence its velocity changes and produces an acceleration. The acceleration is always directed radially toward the centre of the circle. The centripetal acceleration has a magnitude equal to the square of the speed of a body along the curve divided by the distance from the centre of the circle to the moving body. The unit of centripetal acceleration is metre per second square.