Question

A car of mass $ 1000\, kg $ moving with a speed $ 18 \,km \,h^{-1} $ on a smooth road collides with a horizontally mounted spring of spring constant $ 6.25 \times 10^3\, N \,m^{-1} $ . The maximum compression of the spring is

• A. $ 1\,m $
• B. $ 2\,m $
• C. $ 3\,m $
• D. $ 4\,m $

Answer 1

Answer: (B)

Here,
$ m = 1000\, kg $ , $ v = 18\, km \,h^{-1} $
$ = 18 \times\frac{5}{18}ms^{-1} = 5 \,ms^{-1} $
$ k = 6.25 \times 10^{3}\, N\,m^{-1} $
At maximum compression $ x_{m} $ , the kinetic energy of the car is converted entirely into the potential energy of the spring.
$ \therefore \frac{1}{2}mv^{2} = \frac{1}{2}kx^{2}_{m} $
or $ x_{m} = \sqrt{\frac{m}{k}}v $
Substituting the given values, we get
$ x_{m} = \sqrt{\frac{1000\,kg}{6.25 \times 10^{3}\,N\,m^{-1}}} \times5\,ms^{-1} $
$ = 0.4 \times 5 \,m = 2\,m $