Question

A body of mass 5 kg moving with a speed of \[1.5m/s\]on a horizontal smooth surface collides with a nearly weightless spring of force constant \[k = 5N/m\]. The maximum compression of the spring would be :
(A) \[0.5m\]
(B) \[0.15m\]
(C) \[1.5m\]
(D) \[0.12m\]

Answer 1

Hint It is given that a body of specified mass moves with a speed on a smooth horizontal surface undergoes a one dimensional collision with a massless spring. Now, the ball initially undergoes kinetic energy and later this kinetic energy is converted into potential energy of the spring. Calculate maximum compression using this logic.

Complete Step By Step Solution
It is given that a body of mass 5kg is moving with a speed of \[1.5m/s\] on a smooth surface. Thus, there won’t be any friction acting upon the body as it moves along the surface. Now, the total energy undergone by the body, while it is moving from one end to the other in a smooth surface is the kinetic energy of the body. Thus it is given as ,
\[ \Rightarrow \dfrac{1}{2}m{v^2} = KE\]
When this ball strikes a spring of specified stiffness and spring constant, the whole energy is conserved and hence the kinetic energy of the ball is transformed into the potential energy exerted by the spring. Thus according to the law of conservation of energy,
Kinetic energy of the body is equal to the potential energy exerted by the spring. Mathematically,
\[ \Rightarrow \dfrac{1}{2}m{v^2} = \dfrac{1}{2}k{x^2}\]
Cancelling out the common terms we get,
\[ \Rightarrow m{v^2} = k{x^2}\]
On substituting the given values in the equation we get,
\[ \Rightarrow 5{(1.5)^2} = 5{x^2}\]
Cancelling out the common term and cancelling out the square value we get,
\[ \Rightarrow x = 1.5m\]
Hence, the maximum compression of the spring when the ball moving on a smooth surface strikes it would be 1.5 meters.

Thus, option (c) is the right answer for the given question.

Note Law of conservation of energy states that energy can neither be created nor be destroyed. This means that the kinetic and potential energy of an object in a system will be equal to the kinetic and potential energy of another object in the same system, thus preserving the energy.