An F-14 Tomcat fighter jet is taking off from the deck of...

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?

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?

Key Concepts

Finite Difference Method

The finite difference method is a numerical technique used to approximate derivatives by using discrete data points. In kinematics, it means calculating average velocities and accelerations over small time intervals by computing the differences in position and velocity, respectively. This approach is especially useful when continuous data is not available or when measurements have been recorded at discrete time intervals.

Average Velocity

Average velocity is defined as the ratio of the change in position to the change in time over a given interval. It provides a measure of how fast an object is moving in a particular direction over that period, regardless of any variations in speed or direction that may occur within the interval. In discrete data analysis, it is calculated by subtracting the initial position from the final position and dividing the result by the elapsed time.

Average Acceleration

Average acceleration is the ratio of the change in velocity to the change in time over an interval. It quantifies how quickly the velocity of an object changes, giving insight into whether the acceleration is consistent or varying over time. To calculate it using discrete measurements, the difference in velocity between consecutive time intervals is divided by the time difference.

Constant Acceleration

Constant acceleration refers to a situation in which the rate of change of velocity remains uniform over time. When acceleration is constant, the average acceleration calculated over any equal time interval will be the same, and the motion can be predicted using simple kinematic equations. Assessing whether the computed average accelerations are consistent across intervals helps determine if the object is accelerating uniformly.

Transcript

00:02 In this problem, we are told that we have a jet airplane, of which we have measured the position with respect to time when it starts to accelerate or when it starts to fly.

00:18 And on the upper bound here, over here, we have the time.

00:24 And as we see, the time is sampled with 0 .2 seconds distance.

00:29 Distance and then we have the exposition of the plane down there.

00:33 And we have certain values.

00:36 So it's zero at zero seconds.

00:39 0 .7 at 2 .2, 3, 0 .4 and so on.

00:44 And what we want to do here is to calculate the velocity of the plane as a function of time and see whether or not the acceleration can be assumed to be constant during this flight.

00:58 So velocity can be calculated as the change in x over time.

01:06 And what we see then is that we have to calculate x with respect to these two values.

01:12 So then you can say this value here that we calculate the velocity for depends on the two neighboring values.

01:21 Here as well, here, here and so on.

01:25 And as an example then, delta x.

01:27 For the first point here between 0 and 0 .2 seconds is then simply delta x which is 0 .7 over 0 .2 which is 3 .5 meters per second.

01:45 So we're going to write 3 .5 here.

01:48 And you can do this for all values.

01:50 I'm not going to show exactly how calculate all of them, but i will fill in the table here.

01:55 We find that this value here is 11 .5.

02:00 This is 18.

02:02 This is 26.