Position, Displacement, and Average Velocity To set a speed...

Position, Displacement, and Average Velocity 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 $v_{c}$ in a windless situation, should we find the average of $d t_{1}$ and $d / t_{2}$ (method 1) or should we divide $d$ by the average of $t_{1}$ 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 $v_{w}$ to the car's speed $v_{c}$ is 0.0240$?$

Position, Displacement, and Average Velocity
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 $v_{c}$ in a windless situation, should we find the average of $d t_{1}$ and $d / t_{2}$ (method 1) or should we divide $d$ by the average of $t_{1}$ 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 $v_{w}$ to the car's speed $v_{c}$ is 0.0240$?$

Key Concepts

Relative Motion

Relative motion describes how the observed speed of an object is affected by the movement of the medium or another reference frame, such as wind or water currents. In physics, this concept is critical for analyzing situations where an object’s effective speed changes depending on the direction of travel relative to the external flow. Understanding relative motion allows one to separate the inherent speed of an object from the effects introduced by the moving environment.

Appropriate Averaging Methods in Kinematics

When dealing with motion under varying conditions—such as different times for equal distances in opposite directions—choosing the correct method of averaging is essential. The arithmetic mean and the harmonic mean are two different types of averages that apply to different situations. Determining whether to average the speeds directly or to compute the ratio of distance to the average time requires an understanding of how these averaging methods interact with the underlying physics, ensuring that external factors like wind are properly accounted for and eliminated.

Displacement

Displacement is the straight?line change in position of an object from its starting point to its final point. In kinematics, it is distinguished from the total distance traveled, and it is a vector quantity that depends on both magnitude and direction. When analyzing motion—especially with opposing directions—it is essential to consider displacement to accurately derive quantities like average velocity.

Average Velocity

Average velocity is defined as the total displacement divided by the total time taken. This concept is fundamental in kinematics because it provides a measure of the overall change in position without being affected by the path taken. In problems where motion occurs in different directions or under different conditions (like wind), careful consideration of average velocity is required to remove external influences and obtain intrinsic speed values.

Transcript

00:01 We're going to let vw be the velocity of the wind, and then we can say v -c would be the velocity of the car.

00:12 So we can say that first for part a, we can say that during the time interval t -1, they're going to be both moving in the same direction.

00:21 So we can say that the effective speed of the car essentially would be equal to the velocity of the car plus the velocity of the wind, because in this case, the wind is helping the car.

00:38 We can say that the distance traveled would then be d equaling the effective sub 1 times t sub 1, and this would simply be equal to v .c plus v sub w times t sub 1.

00:56 We can then say on the return trip, though, during time interval t sub 2, the car moves in the opposite direction of the wind.

01:05 And so here on the return trip, the effective velocity would be the velocity of the car minus the velocity of the wind.

01:14 Again, the distance traveled would be equal to the effective sub 2 times t sub 2.

01:22 And this would simply be equal to v .c minus v sub w times t sub 2.

01:29 And so we can rewrite the two expressions.

01:32 We can say that v subc plus v sub w would be equal to d over t sub 1.

01:39 And then we can say v subc minus v sub w would be equal to d over t sub 2.

01:46 And so we're going to then add the two equations.

01:49 The velocity of the wind then cancels out...

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