To solve this problem, we'll break it into steps and apply basic principles of kinematics. Since air resistance is ignored, we'll treat it as a simple projectile motion problem.

Step 1: Analyze the vertical motion

We are given:

• The height $h = 2.6 \, \text{m}$,
• The vertical velocity at that point is $v_{\text{vertical}} = -1.5 \, \text{m/s}$ (the negative sign is because the football is descending),
• The initial vertical velocity $v_{0y}$ is unknown,
• The acceleration in the vertical direction is $a_y = -9.8 \, \text{m/s}^2$ (due to gravity).

We can use the following kinematic equation to find the time $t$ from when the football was punted until it reaches this height during its downward descent:

$v_y = v_{0y} + a_y t$

Substitute the known values:

$-1.5 = v_{0y} - 9.8 t$

\tag{1}$$ Next, we apply another kinematic equation for vertical displacement: $$y = v_{0y} t + \frac{1}{2} a_y t^2$$ Substitute the known height $$h = 2.6 \, \text{m}$$: $$2.6 = v_{0y} t - \frac{1}{2} (9.8) t^2$$ Substitute $$v_{0y}$$ from Equation (1): $$2.6 = (9.8 t - 1.5) t - 4.9 t^2$$ Simplify the equation: $$2.6 = 9.8 t^2 - 1.5 t - 4.9 t^2$$ $$2.6 = 4.9 t^2 - 1.5 t$$ Rearrange the terms: $$4.9 t^2 - 1.5 t - 2.6 = 0$$ This is a quadratic equation in the form $$at^2 + bt + c = 0$$, where: - $$a = 4.9$$, - $$b = -1.5$$, - $$c = -2.6$$. ### Step 2: Solve the quadratic equation We can now solve for $$t$$ using the quadratic formula: $$t = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ Substitute the values for $$a$$, $$b$$, and $$c$$: $$t = \frac{-(-1.5) \pm \sqrt{(-1.5)^2 - 4(4.9)(-2.6)}}{2(4.9)}$$ Let's compute this.The two solutions for $$t$$ are: - $$t_1 = 0.897 \, \text{seconds}$$ - $$t_2 = -0.591 \, \text{seconds}$$ (This is not physically meaningful as time cannot be negative.) Thus, the time from when the football was punted until it reaches a height of $$h = 2.6 \, \text{m}$$ during its downward descent is approximately $$t = 0.897 \, \text{seconds}$$. Would you like more details on any part of this solution? ### 5 Related Questions: 1. How can we calculate the maximum height the football reaches? 2. What was the initial vertical velocity of the football? 3. How long was the football in the air before hitting the ground? 4. What was the total horizontal distance traveled by the football? 5. How would the trajectory change if air resistance were considered? ### Tip: For projectile motion problems, always break the motion into horizontal and vertical components to simplify calculations.