To set a speed record in a measured (straight-line) distance...

To set a speed record in a measured (straight-line) distance $d$, a race car must be driven first in one direction (in time $t_{1}$ ) and then in the opposite direction (in time $t_{2}$ ). (a) To eliminate the effects of the wind and obtain the car's speed $v_{e}$ in a windless situation, should we find the average of $d r_{1}$ and $d h_{2}$ (method 1 ) or should we divide $d$ by the average of $l_{1}$ and $t_{2} ?$ (b) What is the fractional difference in the two methods when a steady wind blows along the car's route and the ratio of the wind speed $v_{n}$ to the car's speed $v_{c}$ is $0.0240 ?$

Key Concepts

Fractional Difference (Relative Error) Calculation

This concept deals with quantifying the discrepancy between two methods of calculating speed by expressing the difference as a fraction of one of the values. In the context of rounded-trip measurements with wind, the fractional difference emerges from the ratio between the wind speed and the vehicle’s speed, normally involving a quadratic relationship. This analysis shows how a small wind speed relative to the vehicle’s speed can lead to a quantifiable, yet minor, systematic error in the average speed determined by different methods.

Wind Correction in Kinematic Measurements

Wind can affect time measurements differently on each leg of a journey, altering the effective speeds observed. Understanding how wind influences measured times and speeds is crucial in correcting data to obtain the true speed in still-air conditions. The adjustment often involves algebraic manipulation of the observed speeds or times to account for the wind’s additive or subtractive effects on the vehicle’s motion.

Arithmetic Mean versus Harmonic Mean

When averaging speeds, particularly for round trips of equal distance, it is important to distinguish between the arithmetic mean and the harmonic mean. The arithmetic mean (simply adding the speeds measured in each direction and dividing by two) accurately yields the wind?free speed when the effects of wind are symmetric. In contrast, dividing the distance by the average of the times (which leads to a harmonic mean) can introduce a bias when external factors such as wind are present, causing systematic errors in the determination of the inherent speed.

Average Speed Calculation

This concept involves determining the effective speed of an object over a certain distance. In problems where a vehicle covers the same distance multiple times under different conditions, average speed can be calculated by dividing the total distance by the total time taken. This overall average is a critical measurement in kinematics and is sensitive to variations in speed during segments of the journey.

Transcript

00:03 Hello, in this question we light the speed of the carbivc and we light the speed of the wind in the vw.

00:30 So in order to eliminate the speed of wind, we know that when the wind or the vehicle moves in the same direction as the wind, then the velocity, their relative speed is going to be velocity of calm and the velocity of wind, which is distance over time and in different directions, the velocities will add and that is d over t2 so we can add the two equations in order to eliminate the velocity of the wind and we will get v2 equal to half d over t1 plus well let's write this step first because we will need it so adding the two we have twice velocity of kai equals d1 over t1 plus d2 over t2 and so velocity of card then equals how d1 over t1 plus d2 over t2 so obviously we have used method one now the other method would be to find a velocity v as distance over average time and this can be in the polytap to obtain 2 1 over t 1 over d plus t 2 over t over t now if you have to find a difference let me not close it up you can find a difference between the space calculator is in the two methods let this be our equation a let this be equation b our method 2 so the difference would be a minus b.

04:29 Meta 1 minus method b.

04:30 Speed calculated is imitated 1 and a speed calculated is emitted 2.

04:36 So the difference then comes down to how do over t1 plus d over t2 minus 2, 1 over t1 so this simplifies to d over t1 plus d over t2 minus 4 over t 1 plus t 2 over d.

05:46 This comes down to have 2 vm minus 4, 1 over vc minus v w plus v w plus 1 over vc plus v w plus 2 of wind.

06:25 So we are just applying one in two.

06:32 D over t1 is vc minus vw.

06:37 So t1 over d is the inverse 1 over vc minus vw.

06:40 So these two are being applied to this.

06:43 Sorry, to this.

06:47 You know to give us this...