Question

Use the position function \(s(t)=-16 t^{2}+v_{0} t+s_{0}\) for free-falling objects. A ball is thrown straight down from the top of a 220 -foot building with an initial velocity of -22 feet per second. What is its velocity after 3 seconds? What is its velocity after falling 108 feet?

Short answer

The ball's velocity after 3 seconds is -118 feet per second, and its velocity after falling 108 feet is approximately -81.8 feet per second.

Step-by-step solution

Initialize variables

In our case, we know \(-22\) ft/sec is the initial velocity (\(v_{0}\)), \(220\) ft is the initial position (\(s_{0}\)), and \(16\) ft/sec\(^2\) is the constant of gravity. Thus, our initial formula becomes: \(s(t)=-16t^2-22t+220\).

Calculating the velocity after 3 seconds

To find the velocity of the ball after 3 seconds, we need the derivative of \(s(t)\), that is \(v(t)=-32t-22\), which will give the velocity at time \(t\). Therefore, \(v(3)=-32*3-22=-118\) ft/sec.

Calculating the velocity after falling 108 feet

To find the velocity after the ball has fallen 108 feet, we first need to find the time it takes for the ball to fall that distance. So, we derive an equation for time from \(s(t)=-16t^2-22t+220=112\) (since the ball has 112 feet remaining to fall when it has fallen 108 feet), this gives \(16t^2+22t-112=0\). Solving this quadratic equation and considering only the positive root, we find that \(t\) equals about 1.84 seconds. By substituting that back into the \(v(t)=-32t-22\) equation, we find the velocity at that time to be about \(v(1.84)=-81.8\) ft/sec.

Key concepts

Position Function Calculus

Understanding the position function in calculus is essential when analyzing the motion of free-falling objects. The position function, often denoted as \( s(t) \), represents the location of an object at any given time \( t \). For example, the position function for free-falling objects under Earth's gravity can generally be expressed as \( s(t) = -16t^2 + v_0t + s_0 \), where \( -16 \) represents half the acceleration due to gravity (in feet per second squared), \( v_0 \) is the initial velocity, and \( s_0 \) is the initial position.

The negative sign in front of the 16 indicates that gravity is a downward force in this context. Calculating the position of an object at different times can help us understand its trajectory and forecast its future location. In the exercise provided, by inserting the known values into the position function, we generate a specific equation for the scenario of the ball being thrown downward from a building.

Initial Velocity

Initial velocity, expressed as \( v_0 \), is the velocity at which an object starts its motion. It is a key component in the equations of motion and can greatly influence the trajectory and speed of a moving object. In the context of our exercise, the initial velocity is given as \( -22 \) feet per second, which suggests that the ball is thrown downward with that speed, because the negative sign indicates a direction opposite to the assumed positive upward direction.

It's crucial to consider initial velocity when predicting how long it will take for an object to hit the ground or reach a certain point in its fall, as it adds to or subtracts from the acceleration due to gravity. The impact of initial velocity is clearly seen when we derive the velocity function from the position function, using it to determine the object's speed at various points in time.

Velocity-Time Derivative

The velocity-time derivative is the cornerstone of understanding the relationship between an object’s position and its velocity. In calculus, this derivative is obtained by differentiating the position function, \( s(t) \), with respect to time \( t \). This differentiation gives us the velocity function \( v(t) \), which represents the velocity of the object at any given time.

In the case of our free-falling ball, by taking the derivative of the position function \( s(t) = -16t^2 - 22t + 220 \), we get the velocity function \( v(t) = -32t - 22 \). This tells us the rate at which velocity is changing at any instant. For example, after 3 seconds, we can calculate the velocity by substituting \( t = 3 \) into the velocity function. The deeper understanding of the velocity-time relationship is exemplified in the progression from knowing an object's initial conditions to predicting its speed at any subsequent point in its trajectory.