Problem 6 - Calculus: Early Transcendentals

Question

If a rock is thrown upward on the planet Mars with a velocity of \( 10 m/s \), its height in meters \( t \) seconds after is given by \( y = 10t - 1.86t^2 \). (a) Find the average velocity over the given time intervals: (i) \( [1, 2] \) (ii) \( [1, 1.5] \) (iii) \( [1, 1.1] \) (iv) \( [1, 1.01] \) (v) \( [1, 1.001] \) (b) Estimate the instantaneous velocity when \( t = 1 \).

Step-by-Step Solution

Answer

(a) Average velocities: (i) 4.42 m/s, (ii) 5.35 m/s, (iii) 6.134 m/s, (iv) 7.38 m/s, (v) 7.13 m/s. (b) Instantaneous velocity at \( t = 1 \) is approximately 7 m/s.

Step 1: Understand Formula for Average Velocity

The average velocity over a time interval \([a, b]\) is calculated using the formula \( v_{avg} = \frac{y(b) - y(a)}{b - a} \). This formula finds the change in height divided by the change in time.

Step 2: Calculate Height at Specific Times

For each given time \( t \), calculate the height \( y(t) \) using the formula \( y = 10t - 1.86t^2 \).\(y(1) = 10(1) - 1.86(1)^2 = 10 - 1.86 = 8.14 \)\(y(2) = 10(2) - 1.86(2)^2 = 20 - 7.44 = 12.56 \)Continue calculating as needed for each specific \( t \) value used in subsequent steps.

Step 3: Calculate Average Velocity for Each Interval

Use the heights computed to calculate the average velocity for each specified interval.(i) Interval \([1, 2]\): \[ v_{avg} = \frac{y(2) - y(1)}{2 - 1} = \frac{12.56 - 8.14}{1} = 4.42 \text{ m/s} \](ii) Interval \([1, 1.5]\):\[ y(1.5) = 10(1.5) - 1.86(1.5)^2 = 15 - 4.185 = 10.815 \]\[ v_{avg} = \frac{y(1.5) - y(1)}{1.5 - 1} = \frac{10.815 - 8.14}{0.5} = 5.35 \text{ m/s} \]Repeat this process for intervals (iii), (iv), and (v) using the respective \( t \) values.

Step 4: Estimate Instantaneous Velocity at \( t = 1 \)

As the average velocity is calculated for increasingly smaller intervals around \( t = 1 \), it approaches the instantaneous velocity. Observing the pattern from steps (iii), (iv), and (v), we can approximate the instantaneous velocity at \( t = 1 \).Calculate the average velocities for intervals:(iii) \([1, 1.1]\):\[ y(1.1) = 10(1.1) - 1.86(1.1)^2 = 11 - 2.2466 = 8.7534 \]\[ v_{avg} = \frac{8.7534 - 8.14}{0.1} = 6.134 \text{ m/s} \](iv) \([1, 1.01]\):\[ y(1.01) = 10(1.01) - 1.86(1.01)^2 = 10.1 - 1.8862 = 8.2138 \]\[ v_{avg} = \frac{8.2138 - 8.14}{0.01} = 7.38 \text{ m/s} \](v) \([1, 1.001]\):\[ y(1.001) = 10(1.001) - 1.86(1.001)^2 = 10.01 - 1.86286986 = 8.14713014 \]\[ v_{avg} = \frac{8.14713014 - 8.14}{0.001} = 7.13 \text{ m/s} \]Thus, the instantaneous velocity is approximately 7 m/s.

Key Concepts

Average Velocity

In calculus, average velocity is a concept that helps us understand the motion of an object over a particular interval of time. Imagine a rock being thrown upward on Mars with a certain velocity. To find out how fast it moves on average over several seconds, we use the average velocity formula:

• Average velocity (\( v_{avg} \)) = \( \frac{y(b) - y(a)}{b - a} \)

This formula calculates the change in the height of the rock divided by the time it takes for this change to happen.
For example, if the rock is thrown and we want to know how fast, on average, it travels from 1 to 2 seconds, we look at how the height changes from these two points and divide it by the time interval they encompass.
The average velocity makes it easy to put numbers to how motion varies over selected intervals, whether they are large or becoming infinitesimally small.

Instantaneous Velocity

Unlike average velocity, which looks at broader time intervals, instantaneous velocity zeroes in on a specific moment in time.
Imagine the rock we threw into the Martian sky.
We can think of instantaneous velocity as capturing the rock's speed at a precise moment, like taking a snapshot of its motion.
We estimate this by looking at the average velocities over shorter and shorter intervals around that instant. For example, to find the instantaneous velocity at 1 second after the throw, we compute average velocities for time intervals like from 1 to 1.1 seconds, 1 to 1.01 seconds, and so forth.

• This process helps the average velocity approach the actual (instantaneous) speed at that exact second.

Thus, instantaneous velocity gives us deeper insights into the object's speed at a particular point, beyond just the averages.

Planetary Motion

Planetary motion refers to the study of how planets and other celestial bodies move in space, following certain laws. When we throw a rock upward on Mars, we have a small taste of planetary motion dynamics.

• The equations we use to describe the rock's flight are simplified versions of those that govern planets.

Understanding how objects move due to gravitational forces, especially on different planets, unveils intriguing aspects of physics.
Each planet has unique attributes that influence projectile motion - for Mars, this includes a different atmosphere and gravitational strength.
Studying motion on planets like Mars allows scientists to predict various scenarios, understand gravitational impacts, and satisfy the human curiosity about what lies beyond our Earthly experience.

Quadratic Equations

The motion of our rock on Mars is governed by a quadratic equation, a fundamental concept in algebra. The equation given:

• \( y = 10t - 1.86t^2 \)

This quadratic equation demonstrates how the height of the rock changes over time due to two main effects:

• An initial velocity term (10t), showing how fast the rock would rise if gravity or any other force did not pull it back.
• A gravitational effect, represented by the \(-1.86t^2\), pulling the rock down, slowing its ascent and accelerating its descent.

Quadratic equations like these are pivotal in predicting paths in physics and calculus. They reveal parabolic shapes when plotted, indicating that the rock rises to a peak before falling back, consistent with motion under gravity on any planet.