Two objects with masses of 1.00 kg and 7.00 kg are connected by a light string that passes over a frictionless pulley as in the figure below.

(a) Determine the tension in the string.

(b) Determine the acceleration of each object.

(c) Determine the distance each object will move in the first second of motion if both objects start from rest.

To determine the answers to these questions, we can use Newton's second law of motion and the concept of the tension in a string.

(a) To find the tension in the string, we need to consider the forces acting on each object. The 7.00 kg mass is hanging vertically, so its weight (mg) acts downwards. The tension in the string pulls it upwards. The 1.00 kg mass is on a horizontal surface and has the tension force acting on it in one direction.

Since the pulley is frictionless and the string is light, the tension in the string is the same on both sides. Let's call it T.

For the 7.00 kg object, the net force in the vertical direction is given by:
T - mg = ma₁

For the 1.00 kg object, the net force in the horizontal direction is given by:
T = ma₂

Now, we can solve these two equations simultaneously to find T.

(b) To determine the acceleration of each object, we need to use the equations we derived from Newton's second law.

For the 7.00 kg object:
T - mg = ma₁

For the 1.00 kg object:
T = ma₂

Now, we can solve these two equations simultaneously to find the values of a₁ and a₂.

(c) To determine the distance each object will move in the first second of motion, we can use the equations of motion. Since both objects start from rest, we can use the formula:

Distance = (1/2) * acceleration * time²

For each object, we have their respective accelerations from part (b), and the time is 1 second. We can calculate the distance moved by each object using these values.

By following this method, you can determine the tension in the string, acceleration of each object, and the distance moved by each object in the first second of motion.