We can solve this problem using basic physics formulas related to vertical motion under gravity. Let’s break it down into two parts:

1. Takeoff Speed (Initial Velocity)

When Jade leaps vertically, the only force acting on him is gravity. We can use the following kinematic equation to find his takeoff speed:

$v^2 = u^2 + 2as$

Where:

• $v = 0 \, \text{m/s}$ (since at the peak of his jump, the velocity is 0),
• $u$ is the takeoff speed (initial velocity),
• $a = -9.8 \, \text{m/s}^2$ (acceleration due to gravity),
• $s = 1.29 \, \text{m}$ (the vertical leap).

Rearranging the equation to solve for $u$ (initial velocity):

$0 = u^2 + 2(-9.8)(1.29)$

$u^2 = 2 \times 9.8 \times 1.29$

$u = \sqrt{2 \times 9.8 \times 1.29}$

Let's calculate $u$.

2. Hang Time

The hang time is the total time to reach the peak and return to the ground. The total hang time $T$ can be found by doubling the time to reach the peak (because the time to rise and fall are the same in free-fall motion). The time to reach the peak is given by:

$t = \frac{u}{g}$

Where:

• $u$ is the initial velocity (which we will calculate),
• $g = 9.8 \, \text{m/s}^2$ (acceleration due to gravity).

The total hang time is:

$T = 2t = \frac{2u}{g}$

Let's calculate both the takeoff speed and hang time now.1. Takeoff Speed (Initial Velocity): Jade's takeoff speed is approximately $5.03 \, \text{m/s}$.

2. Hang Time: His total hang time (the time to move upwards to the peak and then return to the ground) is approximately $1.03 \, \text{seconds}$.

Would you like further details or have any questions?

Here are 5 related questions to explore further:

1. How does air resistance affect the hang time in real-life scenarios?
2. What would be Jade’s hang time if he jumped with different gravitational accelerations (e.g., on the Moon)?
3. Can we calculate the maximum height if Jade's hang time is already known?
4. How would the takeoff speed change if Jade’s leap height increased by 20%?
5. How does horizontal motion combine with vertical motion in projectile physics?

Tip: In vertical free-fall motion, the time taken to rise to the peak is always equal to the time taken to fall back down.