Question

which formula can be used to find the tangential speed of an orbiting object?
○ ( v = \frac{2pi r}{t} )
○ ( v = \frac{sqrt{2pi r}}{t} )
○ ( v = g \frac{m_{\text{central}}}{r} )
○ ( v = r sqrt{\frac{m_{\text{central}}}{g}} )

Explanation:

Step1: Recall Tangential Speed Formula

Tangential speed (\(v\)) of an orbiting object (moving in a circular path) is the distance traveled along the circumference per unit time. The circumference of a circle is \(2\pi r\) (where \(r\) is the radius of the orbit), and the time period for one orbit is \(T\). So the formula for tangential speed is \(v=\frac{\text{distance}}{\text{time}}=\frac{2\pi r}{T}\).

Step2: Analyze Other Options

• The second option \(v = \frac{\sqrt{2\pi r}}{T}\) is incorrect as the numerator should be \(2\pi r\) (circumference), not its square - root.
• The third and fourth options involve the gravitational constant \(G\) and masses, which are used for finding orbital speed from gravitational force considerations (centripetal force from gravity), but the basic tangential speed formula from circular motion (distance over time) is \(v=\frac{2\pi r}{T}\).

Answer:

\(v=\frac{2\pi r}{T}\) (the first option among the given choices)