Any offset between a stationary radar gun and a moving...

Any offset between a stationary radar gun and a moving target creates a "cosine effect" that reduces the radar reading by the cosine of the angle between the gun and the vehicle. That is, the radar speed reading is the product of the actual speed and the cosine of the angle. Find the radar readings, to the nearest unit, for Auto A and Auto B shown in the figure. (FIGURE CANNOT COPY) The speed reported by a radar gun is reduced by the cosine of angle $\theta$, shown in the figure, where $r$ represents reduced speed and $a$ represents actual speed. Use the figure to show why this "cosine effect" occurs.

Any offset between a stationary radar gun and a moving target creates a "cosine effect" that reduces the radar reading by the cosine of the angle between the gun and the vehicle. That is, the radar speed reading is the product of the actual speed and the cosine of the angle.
Find the radar readings, to the nearest unit, for Auto A and Auto B shown in the figure.
(FIGURE CANNOT COPY)
The speed reported by a radar gun is reduced by the cosine of angle $\theta$, shown in the figure, where $r$ represents reduced speed and $a$ represents actual speed. Use the figure to show why this "cosine effect" occurs.

Key Concepts

Vector Projection of Velocity

This concept involves decomposing the object's velocity into components along specific directions. The speed measured by a radar is the projection of the true velocity vector onto the line-of-sight direction. Mathematically, this is achieved by multiplying the actual speed by the cosine of the angle between the velocity vector and the radar beam, thus capturing only the effective component along that beam.

Radar Speed Measurement and the Cosine Effect

In radar measurement, the speed recorded is the radial component of the moving object’s velocity. When the object’s path is not aligned directly with the radar’s line-of-sight, the measured speed is reduced. This is because only the component of the velocity vector that points toward or away from the radar is detected, a phenomenon known as the cosine effect.

Trigonometric Cosine Function

The cosine function, a fundamental aspect of trigonometry, relates the angle in a right triangle to the ratio of the adjacent side to the hypotenuse. In the context of radar detection, cosine of the angle between the object’s path and the radar beam serves as a scaling factor, reducing the true speed to the effective speed detected by the radar gun.

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