Question

An object, of mass M is resting on a smooth horizontal surface. A horizontal force F (= F(x)), acts on it such that the acceleration imparted by it to the object, keeps on increasing in direct proportion to the square of the distance (x), moved by the object, from its starting position. The speed of the object, after moving through a distance L, would be

(1) KL √1/2 L (k = constant of proportionality)

(2) KL √3/4 L (k = constant of proportionality)

(3) KL √2/3 L (k = constant of proportionality)

(4) KL √1/6 L (k = constant of proportionality)

Answer 1

(3) KL √2/3 L (k = constant of proportionality)

The acceleration, a (x), imparted to the object, at a distance x from its starting position, is 

a (x) = kx² (k = constant of proportionality) 

The force, F(x), acting on the object, at this position, is 

F(x) = Ma (x) = Mk x² 

∴ Word done in moving a distance dx = dW = F(x) dx

∴ Total work done 

This work increases only the K.E. of the object. Hence if v is the speed of the object, at the position x = L, we have