To calculate the final velocity of a projectile when the range, angle of inclination, and initial velocity are given, you can use the equations of projectile motion. The final velocity refers to the magnitude and direction of the velocity vector at the end of the projectile's trajectory.
Let's denote the range as R, the angle of inclination as θ, the initial velocity as v₀, and the final velocity as v.
The horizontal and vertical components of the initial velocity are given by: v₀x = v₀ * cos(θ) v₀y = v₀ * sin(θ)
Since there is no horizontal acceleration (assuming no air resistance), the horizontal component of velocity remains constant throughout the motion: vₓ = v₀x
The vertical component of velocity changes due to the effect of gravity. The final vertical velocity can be found using the equation: v_y = v₀y - g * t, where g is the acceleration due to gravity (approximately 9.8 m/s²) and t is the time of flight.
The time of flight can be determined using the equation for horizontal displacement: R = vₓ * t, which can be rearranged to: t = R / vₓ.
Substituting this value of t into the equation for the vertical component of velocity, we get: v_y = v₀y - g * (R / vₓ).
The final velocity can be obtained by combining the horizontal and vertical components: v = √(vₓ² + v_y²).
By plugging in the known values of R, θ, and v₀, you can calculate the final velocity v using the above equations.