Problem 26 - Calculus - AP Edition

Question

An object is released from rest (its initial velocity is zero) from the Empire State Building at a height of \(1250 \mathrm{ft}\) above street level (Figure Ex-26 on the next page). The height of the object can be modeled by the position function \(s=f(t)=1250-16 t^{2}.\) (a) Verify that the object is still falling at \(t=5 \mathrm{s}.\) (b) Find the average velocity of the object over the time interval from \(t=5\) to \(t=6 \mathrm{s}.\) (c) Find the object's instantaneous velocity at time \(t=5 \mathrm{s}.\) (Check your book to see figure)

Step-by-Step Solution

Answer

(a) Yes, it is falling. (b) Average velocity is -176 ft/s. (c) Instantaneous velocity is -160 ft/s at \( t=5 \) s.

Step 1: Evaluate Position at t=5s

To check that the object is still falling at time \( t=5 \) s, substitute \( t=5 \) into the position function \( s=f(t)=1250-16t^2 \). Calculate \( f(5) = 1250 - 16(5)^2 = 1250 - 400 = 850 \).

Step 2: Evaluate Position at t=6s

Similarly, substitute \( t=6 \) into the position function to find the position at 6 seconds: \( f(6) = 1250 - 16(6)^2 = 1250 - 576 = 674 \).

Step 3: Verify if Object is Falling

Since the height decreases from 850 ft at \( t=5 \) s to 674 ft at \( t=6 \) s, the object is falling.

Step 4: Calculate Average Velocity

The average velocity over the interval from \( t=5 \) to \( t=6 \) is calculated by the change in position divided by the change in time: \( v_{avg} = \frac{f(6) - f(5)}{6 - 5} = \frac{674 - 850}{6 - 5} = -176 \, \text{ft/s} \).

Step 5: Determine Instantaneous Velocity Formula

The instantaneous velocity is found by differentiating the position function with respect to time: \( v(t) = \frac{d}{dt}[1250 - 16t^2] = -32t \).

Step 6: Calculate Instantaneous Velocity at t=5s

Substitute \( t=5 \) into the velocity formula: \( v(5) = -32(5) = -160 \, \text{ft/s} \).

Key Concepts

Average Velocity

Average velocity is essentially the object's speed over a specific time period. It shows how the position changes as time passes.

• To calculate it, take the difference in the object's positions at the two times.
• Then divide this difference by the total time elapsed.

For instance, consider the object we have. At 5 seconds, it is 850 feet off the ground. Then at 6 seconds, it's 674 feet high.
To find its average velocity between 5 and 6 seconds, do:\[ v_{avg} = \frac{f(6) - f(5)}{6 - 5} = \frac{674 - 850}{6 - 5} = -176 \, \text{ft/s} \]The negative sign indicates that the object is moving downward. So, during this interval, its average speed was 176 ft/s downwards.
Knowing average velocity helps in understanding the overall movement over a time span, even if the speed fluctuated during that time.

Instantaneous Velocity

Imagine you want to know how fast the object is traveling at a precise moment, not over an interval. This is what's called instantaneous velocity.To find this, we take the derivative of the position function. The derivative gives us a function that tells us the speed at any time.

• Our position function is: \( s=f(t)=1250 - 16t^2 \).
• The derivative of this function, with respect to time \( t \), is: \( v(t) = \frac{d}{dt}[1250 - 16t^2] = -32t \).

This new function, \( v(t) = -32t \), is what we call the velocity function.
If you substitute \( t=5 \) into this, you get:\[ v(5) = -32 \times 5 = -160 \, \text{ft/s} \]So, at exactly 5 seconds, the object's speed is 160 ft/s downward. Instantaneous velocity tells us how fast and in what direction the object moves at a particular instant.

Position Function

Position functions model where an object is at any time. They are essential in describing motion because they relate time to position.In our example, the position function is given by:\[ s=f(t)=1250-16t^2 \]Here, the 1250 represents the starting height of the object, in feet, above the ground. The term \(-16t^2\) accounts for the effect of gravity, as it dictates how the object's position changes over time.

• "Position" refers to the location of the object at any time, \( t \).
• The initial height is where the object starts, and as time progresses, the gravity pulls it down.

To determine if the object is falling at a specific time, we can put the time value into this position function. If the position decreases, the object is falling. Understanding the position function allows us to track and predict where the object will be at any given time.