A ball is thrown straight upward with an initial velocity vi and in a time t reaches the top of its flight path, which is a displacement yf above the launch point. Consider that same ball being thrown upward with an initial velocity of 2 x vi x a. Determine the displacement of the top point above the launch path.

To determine the displacement of the ball's top point when thrown with an initial velocity of \( 2 \times v_i \times a \), let's break the problem down step by step using kinematic equations.

### Step 1: Displacement for the initial velocity \( v_i \)

We know that the ball is thrown upward with an initial velocity \( v_i \), and after time \( t \), it reaches the top of its flight path where its velocity becomes zero. The displacement from the launch point to the top of the flight path is given by \( y_f \).

Using the kinematic equation:

\[
v_f^2 = v_i^2 + 2a \Delta y
\]

Where:
- \( v_f = 0 \) (at the top of the flight path),
- \( v_i \) is the initial velocity,
- \( a \) is the acceleration due to gravity, \( -g \),
- \( \Delta y = y_f \) is the displacement.

Substitute \( v_f = 0 \) and \( a = -g \):

\[
0 = v_i^2 - 2g y_f
\]

Solving for \( y_f \):

\[
y_f = \frac{v_i^2}{2g}
\]

### Step 2: Displacement for the new initial velocity \( 2 \times v_i \times a \)

Now, the ball is thrown with a new initial velocity of \( 2 \times v_i \times a \).

Using the same kinematic equation for the new situation:

\[
v_f^2 = (2 v_i a)^2 + 2a \Delta y'
\]

Substitute \( v_f = 0 \) and \( a = -g \):

\[
0 = (2 v_i a)^2 - 2g y_f'
\]

Solving for \( y_f' \), the displacement for the new velocity:

\[
y_f' = \frac{(2 v_i a)^2}{2g}
\]

\[
y_f' = \frac{4 v_i^2 a^2}{2g}
\]

Thus, the new displacement to the top point of the flight path is:

\[
y_f' = 2 a^2 \times \frac{v_i^2}{g}
\]

### Final Answer:

The displacement of the top point above the launch path when the ball is thrown with an initial velocity of \( 2 \times v_i \times a \) is:

\[
y_f' = 2 a^2 \times y_f
\]

This means the new displacement is scaled by \( 2a^2 \) times the original displacement \( y_f \).

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Would you like more details or have any questions?

Here are 5 related questions:
1. What happens to the flight time when the initial velocity is doubled?
2. How does the maximum height change if the initial velocity is tripled?
3. What is the effect of air resistance on the displacement in such a problem?
4. How would you calculate the time to reach the top with the new velocity?
5. How can you express the velocity at any point in the trajectory?

**Tip:** When solving kinematic problems, always check which kinematic equation fits the known quantities like initial velocity, time, and acceleration!

Explore how to calculate the displacement of a ball's top point when thrown with an increased initial velocity. This problem covers kinematic equations and is ideal for students in Grades 9-12, offering a detailed step-by-step solution.

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