Problem 75 - Algebra 1: Concepts and Skills

Question

Use a vertical motion model to find how long it will take for the object to reach the ground. Round your solution to the nearest tenth. You drop your keys from a window 30 feet above ground to your friend below. Your friend does not catch them.

Step-by-Step Solution

Answer

The time it will take for the keys to hit the ground, rounded to the nearest tenth, is approximately 1.4 seconds.

Step 1: Understand the problem

In this exercise, the keys are dropped from a height (h0) of 30 feet with an initial velocity (v0) of 0 (since the keys are simply dropped, not thrown). The aim is to find the time (t) it will take for the keys to reach the ground (i.e., when the height (h) equals 0).

Step 2: Set up the equation

Use the equation for vertical motion, which is `h = -16t^2 + v0t + h0`. Since we know \(v0 = 0\) (the initial velocity is 0 as the keys are simply dropped) and \(h0 = 30\) (the initial height is 30 feet), substitute these values into the equation to get `h = -16t^2 + 30`.

Step 3: Solve the equation

We need to find when the keys hit the ground, which is when \(h = 0\). Set the equation from step 2 to equal 0 and solve for \(t\). The equation becomes `-16t^2 + 30 = 0`. Solving for `t` gives \(t = \sqrt{30/16}\).

Step 4: Round to the nearest tenth

After calculating \(t\), round the result to the nearest tenth for the final answer.

Key Concepts

Quadratic Equations

Quadratic equations are vital in understanding the motion of objects under gravity. They typically have the form \(ax^2 + bx + c = 0\). The solution to such equations can be found using different methods such as factoring, completing the square, or using the quadratic formula: \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\). In real-life scenarios, like the vertical motion model used in the exercise, these equations calculate when an object will reach a certain point. The motion equation \(h = -16t^2 + v_0t + h_0\) here reflects a quadratic equation where:

• \(a = -16\)
• \(b = v_0\) (initial velocity)
• \(c = h_0\) (initial height)

To solve for the time \(t\) that makes the height \(h\) equal zero, substitute any known values and solve the resulting quadratic equation for \(t\). This approach is essential for predicting how long it takes for objects, like the keys, to hit the ground.

Initial Velocity

The initial velocity \(v_0\) is a starting speed of any object in motion. For the problem at hand, the keys are dropped without an initial push, which means their initial velocity is 0. Initial velocity directly impacts the vertical motion. If the keys were thrown downwards or upwards with some force, \(v_0\) would have been a positive or negative number respectively. This starting velocity would change how quickly or slowly the keys reach the ground.
In your calculations, always correctly identify \(v_0\), as it determines how you set up and solve the motion equation. When \(v_0 = 0\), it simplifies our vertical motion model to \(h = -16t^2 + h_0\) which significantly eases the solving process.

Free Fall

Free fall describes objects exclusively under the influence of gravity, with no starting push or resistance from air. In this scenario, the keys are in free fall, meaning that their only driving force is gravity. When objects are dropped from a height, they accelerate downward at a rate of \(32 \, \text{ft/s}^2\) due to Earth's gravity.

• The vertical motion equation reduces to \(h = -16t^2 + h_0\)

Free fall is perfectly modeled by quadratic equations because the relationship between time and height is parabolic. Remember, during free fall, the initial velocity is zero if the object is released without a push. The term \(-16t^2\) reflects the acceleration due to gravity in feet, showing how distance is covered more quickly over time as the object falls.