A satellite in circular orbit fires engines backward (retro-fire). The orbit becomes: <svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 300 260" style="max-width:300px;margin:0.5em auto;display:block;"> <title>Satellite retro-fire — orbit shrinks, fire point becomes apogee</title> <rect width="300" height="260" fill="#faf9f5" rx="6"/> <defs><marker id="ah-q17s4" markerWidth="8" markerHeight="6" refX="8" refY="3" orient="auto"><polygon points="0 0, 8 3, 0 6" fill="#c6613f"/></marker></defs> <!-- Earth --> <circle cx="150" cy="130" r="25" fill="#e8e0d4" stroke="#14140f" stroke-width="1.5"/> <text x="150" y="134" text-anchor="middle" font-size="8" font-family="sans-serif" fill="#14140f">Earth</text> <!-- Original orbit (dashed) --> <circle cx="150" cy="130" r="80" fill="none" stroke="#14140f" stroke-width="1" stroke-dasharray="4,3"/> <!-- Elliptical orbit after retro-fire (shrinks on opposite side) --> <ellipse cx="130" cy="130" rx="100" ry="70" fill="none" stroke="#c6613f" stroke-width="1.5"/> <!-- Fire point (right, now apogee) --> <circle cx="230" cy="130" r="5" fill="#c6613f"/> <text x="240" y="120" font-size="7" font-family="sans-serif" fill="#c6613f">retro-fire</text> <text x="240" y="132" font-size="7" font-family="sans-serif" fill="#c6613f">(apogee)</text> <!-- Retro-fire arrow (backward = downward) --> <line x1="230" y1="138" x2="230" y2="165" stroke="#c6613f" stroke-width="2" marker-end="url(#ah-q17s4)"/> <text x="218" y="160" font-size="7" font-family="sans-serif" fill="#c6613f">v - $\Delta v$</text> <!-- Perigee (left) --> <circle cx="30" cy="130" r="3" fill="#788c5d"/> <text x="30" y="120" text-anchor="middle" font-size="7" font-family="sans-serif" fill="#788c5d">perigee</text> <!-- Legend --> <text x="195" y="238" font-size="8" font-family="sans-serif" fill="#14140f">— — original</text> <text x="195" y="250" font-size="8" font-family="sans-serif" fill="#c6613f">——— after retro-fire</text> </svg>

A uniform sphere has a spherical cavity. The gravitational field inside the cavity is: <svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 280 230" style="max-width:280px;margin:0.5em auto;display:block;"> <title>Uniform sphere with spherical cavity</title> <rect width="280" height="230" fill="#faf9f5" rx="6"/> <defs><marker id="ah-q1s4" markerWidth="8" markerHeight="6" refX="8" refY="3" orient="auto"><polygon points="0 0, 8 3, 0 6" fill="#c6613f"/></marker></defs> <!-- Full sphere --> <circle cx="130" cy="105" r="85" fill="#e8e0d4" stroke="#14140f" stroke-width="1.5"/> <!-- Cavity --> <circle cx="170" cy="105" r="40" fill="#faf9f5" stroke="#c6613f" stroke-width="1.5" stroke-dasharray="4,3"/> <!-- Sphere centre O --> <circle cx="130" cy="105" r="3" fill="#14140f"/> <text x="122" y="120" font-size="8" font-family="sans-serif" fill="#14140f">O</text> <!-- Cavity centre O' --> <circle cx="170" cy="105" r="3" fill="#c6613f"/> <text x="178" y="118" font-size="8" font-family="sans-serif" fill="#c6613f">O'</text> <!-- Uniform field arrows inside cavity --> <line x1="190" y1="85" x2="160" y2="85" stroke="#c6613f" stroke-width="1.5" marker-end="url(#ah-q1s4)"/> <line x1="190" y1="105" x2="160" y2="105" stroke="#c6613f" stroke-width="1.5" marker-end="url(#ah-q1s4)"/> <line x1="190" y1="125" x2="160" y2="125" stroke="#c6613f" stroke-width="1.5" marker-end="url(#ah-q1s4)"/> <!-- E label --> <text x="170" y="145" text-anchor="middle" font-size="9" font-family="sans-serif" fill="#c6613f">E = uniform</text> <!-- superposition note --> <text x="140" y="215" text-anchor="middle" font-size="8" font-family="sans-serif" fill="#14140f">Superposition: full sphere - removed sphere</text> </svg>

A satellite experiences weak air drag. Over time, its speed:

A satellite in circular orbit suddenly has its speed doubled. Its new orbit is:

In a Hohmann transfer from orbit r to 4r, the transfer orbit has semi-major axis: <svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 300 260" style="max-width:300px;margin:0.5em auto;display:block;"> <title>Hohmann transfer orbit from r to 4r</title> <rect width="300" height="260" fill="#faf9f5" rx="6"/> <defs> <marker id="ah-q10s4" markerWidth="8" markerHeight="6" refX="8" refY="3" orient="auto"><polygon points="0 0, 8 3, 0 6" fill="#c6613f"/></marker> <marker id="ah-q10s4b" markerWidth="8" markerHeight="6" refX="8" refY="3" orient="auto"><polygon points="0 0, 8 3, 0 6" fill="#788c5d"/></marker> </defs> <!-- Planet --> <circle cx="150" cy="130" r="18" fill="#e8e0d4" stroke="#14140f" stroke-width="1.5"/> <text x="150" y="134" text-anchor="middle" font-size="8" font-family="sans-serif" fill="#14140f">M</text> <!-- Inner orbit r --> <circle cx="150" cy="130" r="40" fill="none" stroke="#14140f" stroke-width="1" stroke-dasharray="3,3"/> <text x="112" y="98" font-size="7" font-family="sans-serif" fill="#14140f">r</text> <!-- Outer orbit 4r --> <circle cx="150" cy="130" r="105" fill="none" stroke="#14140f" stroke-width="1" stroke-dasharray="3,3"/> <text x="55" y="50" font-size="7" font-family="sans-serif" fill="#14140f">4r</text> <!-- Transfer ellipse (half) --> <path d="M 190,130 A 72.5,72.5 0 0,1 45,130" fill="none" stroke="#c6613f" stroke-width="2"/> <!-- Burn 1 --> <circle cx="190" cy="130" r="4" fill="#c6613f"/> <line x1="190" y1="126" x2="190" y2="100" stroke="#c6613f" stroke-width="2" marker-end="url(#ah-q10s4)"/> <text x="200" y="108" font-size="7" font-family="sans-serif" fill="#c6613f">$\Delta v$_{1}</text> <!-- Burn 2 --> <circle cx="45" cy="130" r="4" fill="#788c5d"/> <line x1="45" y1="134" x2="45" y2="158" stroke="#788c5d" stroke-width="2" marker-end="url(#ah-q10s4b)"/> <text x="28" y="152" font-size="7" font-family="sans-serif" fill="#788c5d">$\Delta v$_{2}</text> <!-- Semi-major axis --> <line x1="45" y1="240" x2="190" y2="240" stroke="#c6613f" stroke-width="1"/> <line x1="45" y1="236" x2="45" y2="244" stroke="#c6613f" stroke-width="1"/> <line x1="190" y1="236" x2="190" y2="244" stroke="#c6613f" stroke-width="1"/> <text x="118" y="253" text-anchor="middle" font-size="8" font-family="sans-serif" fill="#c6613f">a = (r + 4r)/2 = ?</text> </svg>