To solve this problem, we can use the equations of motion under constant acceleration due to gravity. Let's break down the steps.

Given Information:

1. The first ball is thrown upwards with an initial velocity $v$ (we need to find this value).
2. After 1 second, a second ball is thrown upwards with the same initial velocity $v$.
3. The two balls collide 0.5 seconds after the second ball is thrown.
4. Gravitational acceleration $g = 9.8 \, \text{m/s}^2$ (acting downwards).

Steps to Solution:

Let's set up two equations based on the positions of the two balls at the time of collision.

1. Position of the First Ball at the Time of Collision:

   • The time since the first ball was thrown is $1.5$ seconds (since it was thrown 1 second before the second ball, and they collide 0.5 seconds after the second is thrown).
   • Using the formula for the position under uniform acceleration: $y_1 = v \times t - \frac{1}{2} g t^2$ where $t = 1.5$ seconds for the first ball.

   Substituting $t = 1.5$: $y_1 = v \times 1.5 - \frac{1}{2} \times 9.8 \times (1.5)^2$

2. Position of the Second Ball at the Time of Collision:

   • The time since the second ball was thrown is $0.5$ seconds.
   • Using the same formula: $y_2 = v \times t - \frac{1}{2} g t^2$ where $t = 0.5$ seconds for the second ball.

   Substituting $t = 0.5$: $y_2 = v \times 0.5 - \frac{1}{2} \times 9.8 \times (0.5)^2$

3. Setting $y_1 = y_2$ for Collision: Since the balls collide at the same height, we set $y_1 = y_2$: $v \times 1.5 - \frac{1}{2} \times 9.8 \times (1.5)^2 = v \times 0.5 - \frac{1}{2} \times 9.8 \times (0.5)^2$

4. Solve for $v$: Expanding and simplifying this equation will give the value of $v$.

Calculations:

1. Calculate each term and solve for $v$:
   • $1.5v - 0.5 \times 9.8 \times 2.25 = 0.5v - 0.5 \times 9.8 \times 0.25$
   • Simplifying this should yield $v = 9.8 \, \text{m/s}$.

Conclusion:

The initial velocity $v$ with which the balls were tossed is $9.8 \, \text{m/s}$.

Would you like a detailed breakdown of each calculation step or have any further questions?

Here are five related questions to further your understanding:

1. What would happen to the collision time if the second ball was tossed with a higher velocity?
2. How would air resistance impact the outcome of this problem?
3. If the balls were tossed on a different planet with different gravitational acceleration, how would the velocity change?
4. How high would each ball reach before starting to fall if there was no collision?
5. What would the velocities of both balls be at the moment of collision?

Tip: When solving problems involving constant acceleration, breaking down each object's motion with respect to time helps simplify complex interactions, such as collisions.