Question

Use the position func\(\operatorname{tion} s(t)=-4.9 t^{2}+v_{0} t+s_{0}\) for free-falling objects. To estimate the height of a building, a stone is dropped from the top of the building into a pool of water at ground level. How high is the building if the splash is seen 6.8 seconds after the stone is dropped?

Short answer

After calculating, we find that the height of the building (\(s_0\)) is approximately 225.568 meters.

Step-by-step solution

Identify known and unknown variables

Here the known variables are: the acceleration due to gravity represented by -4.9 in the equation, the initial velocity \(v_0\) which is 0 (since the stone was dropped, not thrown), and the time \(t\) which is 6.8 seconds. The unknown variable is the initial position \(s_0\), or the height of the building.

Substitute known variables into the equation

Next, substitute the known variables into the position function. This gives us: \(s(6.8) = -4.9(6.8)^2 + 0(6.8) + s_0\).

Solve for the unknown variable

Rearrange the equation to solve for \(s_0\). This gives: \(s_0 = s(6.8) + 4.9(6.8)^2\). Calculate the right side to find \(s_0\).

Key concepts

Physics in Calculus

Understanding free-fall motion within the context of calculus involves applying principles of differential calculus to the kinematic equations. These equations describe the motion of objects and incorporate time, displacement, velocity, and acceleration. In the context of free-fall, the only force acting on the object is the force of gravity, which accelerates the object at a constant rate. This means that the velocity changes at a constant rate, a concept that is modeled perfectly using calculus.

For instance, the derivative of the position function with respect to time gives us the velocity function, and the derivative of velocity with respect to time provides the acceleration function. With free-fall, since the acceleration due to gravity is constant, the higher-order derivatives after acceleration are zero, making the calculus involved somewhat simpler. By understanding the calculus behind these concepts, students are better equipped to solve problems involving variable motion and to understand the underlying principles that dictate such movements.

Kinematic Equations

Kinematic equations are the bread and butter of solving motion problems in physics. They describe the relationships between position, velocity, acceleration, and time without considering the forces that cause the acceleration. In the case of free-fall motion, these equations become slightly simpler because the acceleration is constant and due to gravity alone.

The general position function for free-fall motion, which is based on the second order differential equation with respect to time, is given by: \( s(t) = -\frac{1}{2}gt^2 + v_0t + s_0 \) where \(g\) represents acceleration due to gravity (-9.8 m/s², though often approximated as -4.9 m/s² to simplify calculations), \(v_0\) is the initial velocity, and \(s_0\) is the initial position. By using kinematic equations, you can predict where an object will be at any given time during its descent, assuming no other forces are acting upon it.

Acceleration Due to Gravity

Acceleration due to gravity is a key component in understanding free-fall motion. On Earth, this value is approximately 9.8 m/s², directed downward towards the center of the planet. However, in many physics problems, it is approximated as 4.9 m/s² when dealing with the squared terms in kinematic equations. This constant acceleration is what causes objects to increase their speed as they fall.

It's important to note that this acceleration is a constant value only because air resistance is typically ignored at the introductory physics level. If air resistance was considered, the motion would become more complex, and different calculations would be required. In free-fall problems like the one in the given exercise, understanding that the acceleration due to gravity is constant allows us to predict the motion of falling objects with high accuracy.

Initial Velocity

Initial velocity, represented by \(v_0\) in kinematic equations, is the speed at which an object begins its motion. In scenarios where the object is dropped—as opposed to being thrown or shot—its initial velocity is 0. This is a crucial factor in solving free-fall problems.

Understanding the role of initial velocity in motion problems allows us to establish the starting point for our calculations. A nonzero initial velocity would affect the object's motion and, consequently, the outcomes of the kinematic equations. In the exercise, since the stone was dropped, the initial velocity is 0, simplifying the equation and allowing us to easily solve for the height of the building by focusing purely on the effects of gravity over time on the stone.