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A body of mass m is projected vertically upward from the Earth's surface with speed lambda*vₑ, where vₑ is the escape speed and lambda < 1 (air resistance is neglected). Find the maximum distance from the centre of the Earth that the body can reach. (R = radius of Earth)
- R / (1 + lambda²)
- R / (1 - lambda²)
- R / (1 - lambda)
- lambda²*R / (1 - lambda²)
Correct answer: R / (1 - lambda²)
Solution
vₑ = sqrt(2GM/R) => vₑ² = 2GM/R. Initial KE = (1/2)m*(lambda*vₑ)² = lambda²*GMm/R. Initial PE = -GMm/R. Total energy E = GMm/R*(lambda² - 1). At max height r_max: KE=0, PE = -GMm/r_max. Energy conservation: -GMm/r_max = GMm(lambda²-1)/R => 1/r_max = (1-lambda²)/R => r_max = R/(1-lambda²).
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