Ten days after it was launched toward Mars in December 1998, the Mars Climate Orbiter spacecraft (mass 629 kg) was 2.87 × 10⁶ km from the earth and traveling at 1.20 × 10⁴ km/h relative to the earth. At this time, what were (a) the spacecraft's kinetic energy relative to the earth and (b) the potential energy of the earth–spacecraft system?

Verified step by step guidance

1. To find the spacecraft's kinetic energy relative to the Earth, use the kinetic energy formula: \( KE = \frac{1}{2}mv^2 \), where \( m \) is the mass of the spacecraft and \( v \) is its velocity. First, convert the velocity from km/h to m/s by multiplying by \( \frac{1000}{3600} \).
2. Substitute the mass of the spacecraft (629 kg) and the converted velocity into the kinetic energy formula to calculate the kinetic energy.
3. For the potential energy of the Earth-spacecraft system, use the gravitational potential energy formula: \( U = -\frac{G M m}{r} \), where \( G \) is the gravitational constant \( 6.674 \times 10^{-11} \text{Nm}^2/\text{kg}^2 \), \( M \) is the mass of the Earth \( 5.972 \times 10^{24} \text{kg} \), \( m \) is the mass of the spacecraft, and \( r \) is the distance between the Earth and the spacecraft.
4. Convert the distance from km to meters by multiplying by 1000. Substitute the values of \( G \), \( M \), \( m \), and \( r \) into the potential energy formula to calculate the potential energy.
5. Ensure all units are consistent (mass in kg, distance in meters, velocity in m/s) and perform the calculations to find the kinetic and potential energies.

Key Concepts

Kinetic Energy

Kinetic energy is the energy an object possesses due to its motion, calculated using the formula KE = 0.5 * m * v^2, where m is the mass and v is the velocity of the object. For the Mars Climate Orbiter, this involves converting its velocity from km/h to m/s and using its mass to find the kinetic energy relative to Earth.

Gravitational Potential Energy

Gravitational potential energy in a two-body system, like the Earth and the spacecraft, is given by U = -G * (m1 * m2) / r, where G is the gravitational constant, m1 and m2 are the masses of the two bodies, and r is the distance between their centers. This concept helps determine the potential energy of the Earth-spacecraft system based on their separation.

Relative Motion

Relative motion refers to the calculation of an object's motion as observed from a particular frame of reference. In this problem, the spacecraft's velocity and energy are considered relative to Earth, meaning calculations must account for the spacecraft's speed and position as observed from Earth, not an absolute frame.