Question

Blocks A and B are at rest with the spring at its natural length and block C collides elastically with A. Then,

• A. The kinetic energy of the A - B system at maximum compression of the spring is zero.
• B. The kinetic energy of the A - B system at maximum compression of the spring is $100\text{J}$.
• C. The maximum compression of the spring is $1.5\text{m}$.
• D. The maximum compression of the spring is $1\text{m}$.

Answer 1

Answer: (B), (D)

The correct options are
B The kinetic energy of the A - B system at maximum compression of the spring is $100\text{J}$.
D The maximum compression of the spring is $1\text{m}$.

From the data given in the question,
Block C moving with velocity $20\text{m/s}$ upon colliding with A elastically, will come to rest instantly and block A will start moving with $20\text{m/s}$.
At maximum compression, blocks A and B will move with a common velocity equal to the velocity of their centre of mass as shown in the figure below.
Let us suppose, the velocity of COM be $v_0$ and maximum compression of the spring be $x$.
According to question,


From figure II:
$v_{COM}=v_0=\frac{1\times 20+1\times 0}{1+1}=10\text{m/s}---\left(1\right)$
$[\because v_{COM}=\frac{m_1{v}_1+m_2{v}_2}{m_1+m_2}]$
We know that in collision, only internal forces operate and hence $\sum F_{int}=0$. Therefore, there is no net force on system A.B.
Thus, $V_{COM}$ will not change.

On applying law of conservation of mechanical energy between blocks A and B as shown in figure II and III,
$M.E_i=M.E_f$
$\Rightarrow M.E_{II}=M.E_{III}$
Initial velocity of Block A $\left(u_2\right)=20\text{m/s}$
Initial velocity of Block B $\left(u_3\right)=0\text{m/s}$
Final velocity of Block A $\left(v_2\right)=10\text{m/s}$
Final velocity of Block B $\left(v_3\right)=10\text{m/s}$
$\frac{1}{2}m{u}_2^2+0=\frac{1}{2}m{v}_2^2+\frac{1}{2}m{v}_3^2+\frac{1}{2}k{x}_0^2$
$\Rightarrow \frac{1}{2}\times 1\times {20}^2+0=\frac{1}{2}\times 1\times {10}^2+\frac{1}{2}\times 1\times {10}^2+\frac{1}{2}\times 200\times x_0^2$
$\Rightarrow x_0=1\text{m}$ is the maximum compression in the spring.

Kinetic energy of A - B system at maximum compression.
$=\frac{1}{2}\times 1\times {10}^2+\frac{1}{2}\times 1\times {10}^2$
$K.E=100\text{J}$
Options B and D are correct.