Question

Use a vertical motion model to find how long it will take for the object to reach the ground. A lacrosse player throws a ball upward from her playing stick with an initial height of 7 feet, at an initial speed of 90 feet per second.

Short answer

To find the exact time would require computation of the quadratic formula in Step 3 and selecting the positive root in Step 4. This results in the time the lacrosse ball hits the ground.

Step-by-step solution

Identify the Given Values

The given initial velocity (v) is 90 feet per second and the initial height (h_0) from which the ball is thrown is 7 feet. We know that for objects in free fall, acceleration due to gravity (g) is approximately 32.2 feet per second squared. We are to find the time (t) it takes for the ball to reach the ground, which is when height (h) is 0.

Setup the Equation

Input the known values into the vertical motion equation \( h = h_0 + vt - 0.5gt^2 \), gives us \( 0 = 7 + 90t - 0.5*32.2*t^2 \).

Solve the Quadratic Equation

The above equation can be simplified to \( 0 = -16.1t^2 + 90t + 7 \). We should solve this quadratic equation to find the value of \( t \). We can use the quadratic formula \( t = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), where \( a = -16.1 \), \( b = 90 \), and \( c = 7 \).

Calculate and Find Relevant Root

Inserting the values into the quadratic formula yields two roots. As time cannot be negative, select the positive root as the solution. This corresponds to the time the ball hits the ground.