A satellite circles a planet at radius 8.00 × 10⁶ m with speed 6.00 × 10³ m s⁻¹. Determine the planet's gravitational parameter GM, orbital period, gravitational field strength and specific total energy. Find escape speed from that radius and the tangential speed increase required. Explain how the field, potential and total energy signs differ, taking zero potential at infinity.

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GM = 2.88 × 10¹⁴ m³ s⁻²; period = 8.38 × 10³ s; field = 4.50 N kg⁻¹; specific energy = −1.80 × 10⁷ J kg⁻¹; escape speed = 8.49 km s⁻¹ and Δv = 2.49 km s⁻¹. Circular motion requires GM/r² = v²/r, so GM = v²r = (6.00 × 10³)²(8.00 × 10⁶) = 2.88 × 10¹⁴ m³ s⁻². Period is circumference divided by speed: T = 2πr/v = 2π(8.00 × 10⁶)/(6.00 × 10³) = 8.38 × 10³ s. Field strength is GM/r² = v²/r = 4.50 N kg⁻¹ toward the planet. Gravitational potential is −GM/r = −v². Specific total energy in a circular orbit is −GM/(2r) = −v²/2 = −1.80 × 10⁷ J kg⁻¹. Escape requires zero total energy, so v_esc = √(2GM/r) = √2v = 8.49 km s⁻¹ and Δv = 2.49 km s⁻¹. Field strength is a positive magnitude with inward direction; potential and bound-orbit total energy are negative relative to infinity.

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