Problem 31 - Algebra 2

Question

For Exercises 30 and \(31,\) use the formula \(h(t)=v_{0} t-16 t^{2}\) where \(h(t)\) is the height of an object in feet, \(v_{0}\) is the object's initial velocity in feet per second, and \(t\) is the time in seconds. BOATING A boat in distress launches a flare straight up with a velocity of 190 feet per second. Ignoring the height of the boat, how many seconds will it take for the flare to hit the water?

Step-by-Step Solution

Answer

The flare hits the water after approximately 11.875 seconds.

Step 1: Understand the Formula

The formula given is \(h(t) = v_{0} t - 16 t^{2}\), where \(h(t)\) represents the height, \(v_{0}\) is the initial velocity, and \(t\) is the time in seconds. We need to find the time \(t\) when the flare hits the water, which corresponds to \(h(t) = 0\).

Step 2: Substitute Known Values

We know the initial velocity \(v_{0} = 190\) feet per second. Thus, substitute into the formula: \(0 = 190t - 16t^2\).

Step 3: Rearrange the Equation

Move all terms to one side of the equation to set it to 0: \(16t^2 - 190t = 0\).

Step 4: Factor the Quadratic Equation

Factor out the common factor \(t\): \(t(16t - 190) = 0\). This gives us two possible solutions: \(t = 0\) or \(16t - 190 = 0\).

Step 5: Solve for t when not zero

Solve \(16t - 190 = 0\). Add 190 to both sides: \(16t = 190\). Then divide by 16: \(t = \frac{190}{16}\).

Step 6: Simplify the Solution

Calculate \(\frac{190}{16}\) to find \(t\). Simplifying gives \(t \approx 11.875\) seconds.

Key Concepts

Understanding Initial Velocity

Initial velocity, often denoted as \(v_{0}\), is the speed at which an object begins its motion. In many physics problems involving projectiles, such as the one about a flare launched from a boat, initial velocity is crucial.

• Initial velocity determines how high and how far an object will go before gravity slows it down and makes it fall back.
• Mathematically, it's part of the formula that describes an object's height over time when affected by gravity: \(h(t) = v_{0} t - 16 t^{2}\). This formula shows that initial velocity contributes positively to the height decreases as time increases.

Knowing the initial velocity helps us predict how the object moves. For example, a higher initial velocity means the flare would stay in the air longer.

Height of an Object

The height of an object in motion, denoted by \(h(t)\), is how far above the ground the object is at any given time \(t\). When dealing with projectiles, understanding their height at different times helps us grasp the dynamics of their motion.

• In the formula \(h(t) = v_{0} t - 16 t^{2}\), \(h(t)\) represents the height in feet.
• As \(t\) increases, the height first increases until the object reaches its peak, then decreases as gravity pulls it down.

The goal of such equations is often to find specific points in time when \(h(t)\) reaches certain values, like when the flare returns to the ground, meaning \(h(t) = 0\).

Time of Flight

The time of flight refers to the duration for which an object remains in motion above the ground from launch to landing. Calculating the time of flight is essential in problems involving projectile motion, like when determining how long it takes for a flare to reach the water.

• To find this time, you set the height equation \(h(t) = v_{0} t - 16 t^{2}\) equal to zero to find when the object is at ground level.
• In this case, the equation \(0 = 190t - 16t^2\) is solved for \(t\) to find when the flare will hit the water.

Solving this equation correctly lets you know the precise moment the object completes its journey.

Factoring Quadratic Equations

Factoring quadratic equations is a fundamental skill in solving projectile motion problems and many other mathematical situations.

• The equation \(16t^2 - 190t = 0\) must be factored to find the values of \(t\) where the object is at ground level.
• This involves taking out the common factor \(t\), giving us \(t(16t - 190) = 0\).

Solving the resultant equation \(t(16t - 190) = 0\) gives potential solutions for \(t\). Factoring helps break down complex expressions into simpler parts, making it easier to find meaningful solutions like when the flare hits the water at approximately 11.875 seconds.