Imagine two protons traveling past each other at a distance \(d,\) with relative speed 0.9c. Compared with two stationary protons a distance \(d\) apart, the gravitational force between these two protons will be a. smaller, because they interact for less time. b. smaller, because the moving proton acts as if it has less mass. c. the same, because the particles have the same mass. d. larger, because the moving proton acts as if it has more mass.

Understanding the gravitational force between two objects is a fundamental aspect of physics. Newton's Law of Gravitation helps us determine this force. The equation is expressed as follows: \ \[ F = G \frac{m_1 m_2}{d^2}\ \] where:

• \text{F} is the gravitational force between two masses.
• \text{G} is the gravitational constant (\approx 6.674 \times 10^{-11} \text{N} \text{m}^{2} \text{kg}^{-2}).
• \text{m}_1 \text{and m}_2 are the masses of the two objects.
• \text{d} is the distance between the centers of the two objects.

This law states that the force of gravity is directly proportional to the product of the two masses and inversely proportional to the square of the distance between them. Therefore, if either of the masses increases, so does the gravitational force. Conversely, if the distance increases, the force decreases.

Relativistic mass is a crucial concept in Einstein's theory of relativity. It describes how an object's mass increases with its velocity. The equation for relativistic mass (\text{m}_r) is: \ \ \[ m_r = \frac{m_0}{\text{\sqrt{1 - (v^2/c^2)}}} \ \ \]
Here,

• \text{m}_0 is the rest mass of the object.
• \text{v} is the velocity of the object.
• \text{c} is the speed of light (\approx 3 \times 10^8 \text{m/s}).

As an object's velocity approaches the speed of light, its relativistic mass increases significantly. At velocities much less than the speed of light, this increase is negligible. However, at speeds close to \text{c}, the relativistic effects become apparent.
In our exercise, protons traveling at 0.9 \text{c} have a much higher relativistic mass than when they are stationary. This impacts the gravitational force between them, as explained by the previous section.

The velocity of an object has a direct impact on its mass due to relativistic effects. As the speed increases, so does the mass of the object. This can be visualized using the relativistic mass formula: \ \ \[ m_r = \frac{m_0}{ \sqrt{1 - ( v^2 / c^2 )} } \ \ \]
As velocity approaches the speed of light:

• The denominator of the fraction (\text{1 - (\text{v}^2/\text{c}^2)}) approaches zero, causing the overall value (relativistic mass \text{m}_r) to become very large.
• When \text{v} is 0.9 \text{c}, the equation shows a substantial increase in mass.

This increase in mass results in a stronger gravitational force between the fast-moving protons compared to stationary protons. The key takeaway is that at high velocities approaching the speed of light, an object's mass significantly increases, leading to greater gravitational attractions when applying Newton's Law of Gravitation.