$A$ girl riding a bicycle along a straight road with a speed of $5 \, m/s$ throws a stone of mass $0.5 \, kg$ which has a speed of $15 \, m/s$ with respect to the ground along her direction of motion. The mass of the girl and bicycle is $50 \, kg$. Does the speed of the bicycle change after the stone is thrown? What is the change in speed,if so?

(A) Initial mass of the system (girl + bicycle + stone) $= 50 \, kg + 0.5 \, kg = 50.5 \, kg$.
Initial velocity of the system $u = 5 \, m/s$.
Initial momentum $P_i = (50.5 \, kg) \times (5 \, m/s) = 252.5 \, kg \cdot m/s$.
After throwing the stone,the mass of the stone is $m_s = 0.5 \, kg$ and its velocity is $v_s = 15 \, m/s$.
Let the new velocity of the girl and bicycle be $v_b$. The mass of the girl and bicycle is $m_b = 50 \, kg$.
By the law of conservation of linear momentum,$P_i = P_f$.
$252.5 = (m_s \times v_s) + (m_b \times v_b)$.
$252.5 = (0.5 \times 15) + (50 \times v_b)$.
$252.5 = 7.5 + 50 \times v_b$.
$50 \times v_b = 245$.
$v_b = 4.9 \, m/s$.
Yes,the speed of the bicycle changes. The change in speed is $5 \, m/s - 4.9 \, m/s = 0.1 \, m/s$.