A

To find the speed of the shadow of the particle on the wall, we can use the relationship between the angle of the shadow and the motion of the particle.
Step 1: Define the variables:
Let \( R \) be the radius of the circular path, \( heta \) be the angle subtended by the particle and the center, and velocity of the particle be \( v \).
Step 2: Use trigonometry to find the distance of the shadow from the center: The distance \( D \) from the center to the shadow on the wall can be expressed as \( D = R \sec{\theta} \).
Step 3: Differentiate this distance with respect to time to find the speed of the shadow: \( rac{dD}{dt} = R \sec{\theta} an{\theta} \frac{d\theta}{dt} \).
Here, \( \frac{d\theta}{dt} = \frac{v}{R} \).
Step 4: Substitute this into the equation to find the speed of the shadow:
\( \frac{dD}{dt} = R \sec{\theta} \tan{\theta} \left(\frac{v}{R}\right) = v \sec{\theta} \tan{\theta} \).
Therefore, the formula for the speed of the shadow is given by
\( v_{shadow} = v \sec{\theta} \tan{\theta} \). This indicates how the speed of the shadow is related to the speed of the particle and the angle of projection.
Therefore, the speed of the shadow at the instant shown in the figure is correctly represented by option A.