Final answer:
The total velocity of the vehicle is 7.8 m/s at an angle of 63.4 degrees with the vertical, determined using the Pythagorean theorem for the magnitude and arctangent function for the direction angle.
Step-by-step explanation:
To determine the magnitude and direction of the total velocity of the descent vehicle landing on Mars, we can use the Pythagorean theorem for the magnitude, and trigonometric functions for the direction. Since the problem mentions a vertical velocity of 7.0 m/s downward and a horizontal velocity of 3.5 m/s, we treat these as perpendicular components of the total velocity vector.
The magnitude (V) can be calculated as:
V = √(v_vertical² + v_horizontal²)
V = √(7.0 m/s)² + (3.5 m/s)²)
V = √(49 + 12.25)
V = √(61.25 m²/s²)
V ≈ 7.8 m/s
The direction (θ) can be found using the arctangent function:
θ = tan⁻¹(v_horizontal / v_vertical)
θ = tan⁻¹(3.5 m/s / 7.0 m/s)
θ ≈ 26.6 degrees with the vertical
However, the answer choices are listed in terms of the angle with either the horizontal or the vertical. In this case, to convert to the angle with the vertical, we subtract this angle from 90 degrees:
90 - θ = 90 - 26.6 = 63.4 degrees with the vertical
The correct answer is thus 7.8 m/s at an angle of 63.4 degrees with the vertical.