a) Well, Sphere 1 seems to be a real go-getter. After all that swinging and colliding, its velocity is really up in the air...literally! But let's calculate it anyway. First, we need to find the potential energy that Sphere 1 gained when it was pulled to the left. The formula for potential energy is mgh, where m is the mass (30 g), g is the acceleration due to gravity (you know, that thing that keeps us grounded), and h is the height (8.0 cm). Convert 30 g to kilograms by dividing it by 1000, then multiply by 9.8 m/s^2 for g, and multiply by 0.08 m for h. Now you have the potential energy. But since energy is always conserved, this potential energy will be converted into kinetic energy just after the collision. And the formula for kinetic energy is (1/2)mv^2, where m is the mass (30 g again), and v is the velocity. So, equating the potential energy to the kinetic energy, you can solve for v. Go ahead and crunch those numbers, smarty pants!
b) Now that Sphere 1 has had its fun colliding, it's time to see how high it can swing to the left. Remember all that potential energy we calculated earlier? Well, now it's going to be converted back into kinetic energy as Sphere 1 swings up. And since energy is a bit of a show-off, it stays constant throughout this reversal of fortune. So, using the same formula as before (1/2)mv^2, you can find the velocity of Sphere 1 just before it reaches its maximum height. Then plug that into the formula for potential energy mgh, where m is the mass (30 g), g is the acceleration due to gravity (9.8 m/s^2), and h is the maximum height you need to find. Solve for h, my friend. It's time for Sphere 1 to reach new heights!
c) Now it's time to talk about Sphere 2. Poor thing, it had a collision with Sphere 1 and now it's feeling a bit dizzy. To find its velocity just after the collision, you can use the equation for conservation of momentum. The sum of the initial momenta (which is just m1v1 + m2v2, where m1 is the mass of Sphere 1, v1 is its initial velocity just before the collision, m2 is the mass of Sphere 2, and v2 is its initial velocity just before the collision) should equal the sum of the final momenta (which is m1v1' + m2v2', where v1' is the final velocity of Sphere 1 just after the collision, and v2' is the final velocity of Sphere 2 just after the collision). In an elastic collision, like this one, momentum is conserved. So, you can solve for v2' using this equation. Get ready to put your thinking cap on and find the velocity of Sphere 2 after the collision.