Two objects with masses of $9.00 \mathrm{~kg}$ and $25.00 \mathrm{~kg}$ are connected by a light string that passes over a frictionless pulley. Determine (a) the tension in the string, (b) the acceleration of each object, (c) the distance each object will move in the first second of motion if both objects start from rest.

Step 1 ：Given two objects with masses of 9.00 kg and 25.00 kg are connected by a light string that passes over a frictionless pulley.

Step 2 ：For the 9 kg object, the forces acting on it are the tension T in the string and the force of gravity (9*9.8 N). Since the object is accelerating upwards, we can write the equation as \(T - 9*9.8 = 9*a\), where a is the acceleration.

Step 3 ：For the 25 kg object, the forces acting on it are the tension T in the string and the force of gravity (25*9.8 N). Since the object is accelerating downwards, we can write the equation as \(25*9.8 - T = 25*a\).

Step 4 ：We can solve these two equations simultaneously to find the values of T and a. The solution is \(T = 129.71 N\) and \(a = 4.61 m/s^2\).

Step 5 ：Once we have the acceleration, we can use the equation of motion to find the distance each object will move in the first second. The equation of motion is \(d = 0.5*a*t^2\), where d is the distance, a is the acceleration, and t is the time. Since the objects start from rest, the initial velocity is zero, so this equation simplifies to \(d = 0.5*a*t^2\).

Step 6 ：Substituting the values of a and t into the equation, we get \(d = 2.31 m\).

Step 7 ：Final Answer: The tension in the string is approximately \(\boxed{129.71 N}\), the acceleration of each object is approximately \(\boxed{4.61 m/s^2}\), and the distance each object will move in the first second of motion is approximately \(\boxed{2.31 m}\).