Question

An F-14 Tomcat fighter jet is taking off from the deck of the USS Nimitz aircraft carrier with the assistance of a steam-powered catapult. The jet's location along the flight deck is measured at intervals of \(0.20 \mathrm{~s} .\) These measurements are tabulated as follows: $$ \begin{array}{|l|l|l|l|l|l|l|l|l|l|l|l|} \hline t(\mathrm{~s}) & 0.00 & 0.20 & 0.40 & 0.60 & 0.80 & 1.00 & 1.20 & 1.40 & 1.60 & 1.80 & 2.00 \\ \hline x(\mathrm{~m}) & 0.0 & 0.70 & 3.0 & 6.6 & 11.8 & 18.5 & 26.6 & 36.2 & 47.3 & 59.9 & 73.9 \\ \hline \end{array} $$ Use difference formulas to calculate the jet's average velocity and average acceleration for each time interval. After completing this analysis, can you say if the F- 14 Tomcat accelerated with approximately constant acceleration?

Short answer

Answer: We can determine if the F-14 Tomcat accelerated with approximately constant acceleration by analyzing the calculated average accelerations for each time interval. If the average acceleration values remain roughly constant throughout the measurements, we can conclude that the F-14 Tomcat accelerated with approximately constant acceleration.

Step-by-step solution

Calculate the average velocity for each time interval

To find average velocity, we will use the following formula: $$ v_{avg} = \frac{\Delta x}{\Delta t} $$ Calculate the average velocity for each time interval by dividing the difference in position (\(\Delta x\)) by the time interval (\(\Delta t = 0.20 \mathrm{~s}\)).

Calculate the average acceleration for each time interval

To find average acceleration, we will use the following formula: $$ a_{avg} = \frac{\Delta v_{avg}}{\Delta t} $$ Calculate the average acceleration for each time interval by dividing the difference in average velocity (\(\Delta v_{avg}\)) by the time interval (\(\Delta t = 0.20 \mathrm{~s}\)).

Analyze the average accelerations for constant acceleration

Observing the calculated average accelerations, determine if they are approximately constant throughout the measurements. If the average acceleration remains roughly constant, we can conclude that the F-14 Tomcat accelerated with approximately constant acceleration.

Key concepts

Average Velocity

Understanding the concept of average velocity is crucial for studying kinematics, the branch of physics that deals with motion without considering the forces that cause it. By definition, average velocity is the total displacement divided by the total time taken to cover that displacement.

Let's consider an F-14 Tomcat fighter jet taking off; to calculate its average velocity over a specific time interval, we use the formula: \[ v_{avg} = \frac{\Delta x}{\Delta t} \]In this context, \(\Delta x\) represents the change in the jet's position on the flight deck, while \(\Delta t\) is the duration of the time interval. For instance, if the jet moves from 0 meters to 73.9 meters in 2 seconds, its average velocity would be \[ v_{avg} = \frac{73.9 \mathrm{~m} - 0 \mathrm{~m}}{2 \mathrm{~s} - 0 \mathrm{~s}} = 36.95 \frac{m}{s} \]This calculation would be the first step in analyzing the jet's movement over the time period outlined.

Average Acceleration

The next fundamental concept in kinematics is average acceleration, which tells us how quickly the velocity of an object changes over time. The formal definition states it as the change in velocity divided by the time over which the change occurs.

In our F-14 Tomcat example, we compute average acceleration for each time interval using the formula: \[ a_{avg} = \frac{\Delta v_{avg}}{\Delta t} \]Here, \(\Delta v_{avg}\) is the difference in average velocity between two consecutive measurements, and as before, \(\Delta t\) is the time interval. To illustrate, if in the first interval the jet's velocity changes from 0 to 36.95 m/s over 2 seconds, the average acceleration is:\[ a_{avg} = \frac{36.95 \mathrm{~m/s} - 0 \mathrm{~m/s}}{2 \mathrm{~s}} = 18.475 \frac{m}{s^2} \]By calculating average acceleration at different intervals, we gain insight into the jet's rate of velocity change throughout its takeoff.

Constant Acceleration

Continuous uniform acceleration is a particular case where an object's average acceleration remains constant throughout its motion. This is essential in simplifying the equations of motion and is a common assumption in kinematic problems because it allows us to use straightforward formulas to predict future motion.

Referring to our jet, we would expect to find a roughly constant average acceleration to say it has uniform acceleration. By evaluating the calculated average accelerations for each time interval and finding them to be approximately equal, we could conclude the jet indeed has constant acceleration. However, in practical scenarios, slight variations might occur due to factors such as air resistance and mechanical imperfections. Still, for many cases in kinematics, assuming a constant acceleration can be a very close approximation of the true motion of an object.