1.(a) Explain the term time dilation.

(b) Show that an interval of time observed in a moving frame of reference will be less than the same interval of the observed in a stationary frame of reference.
2. A rocket was found to be 150m when measured on earth. It then leaves and move at a constant velocity of 2 x 10^8m/s. What is its new length?
3. Show that the force acting on a particle as observed by two observers in the two inertial frames if reference is the same. Given v<<c.
4. Two spaceships A and B are moving in opposite directions A with a speed of 0.9c while B with a speed of 0.6c. Find the relative velocity of B with respect to A.
5.A satellite goes round the earth in 90 minutes in a circular orbit. Calculate the height of the satellite above the earth taking the earth to be a sphere of radius 6370km. The value of g at the orbit of satellite is 9.8m/s^2

1. (a) Time dilation refers to the phenomenon in which time is experienced differently by observers in different frames of reference. It is a consequence of the theory of relativity, specifically the theory of special relativity proposed by Albert Einstein. According to this theory, the relative motion between two observers will cause a difference in the time intervals they measure.

(b) To show that an interval of time observed in a moving frame of reference will be less than the same interval observed in a stationary frame of reference, we can consider the time dilation formula derived from special relativity:

Δt' = Δt / √(1 - (v^2/c^2))

Where:
Δt' is the observed time interval in the moving frame of reference,
Δt is the observed time interval in the stationary frame of reference,
v is the relative velocity between the two frames of reference, and
c is the speed of light in a vacuum.

When v is non-zero, the term (v^2/c^2) is less than 1. As a result, the denominator √(1 - (v^2/c^2)) is greater than 1. Therefore, Δt' < Δt, indicating that the observed time interval in the moving frame of reference is shorter than the same interval observed in the stationary frame of reference.

2. To determine the new length of the rocket, we can apply the concept of length contraction, which is another implication of the theory of relativity. According to length contraction, objects appear to be shorter along the direction of their motion when observed from a reference frame in relative motion.

The formula for length contraction is given by:

L' = L * √(1 - (v^2/c^2))

Where:
L' is the length observed in the moving frame of reference,
L is the original length of the rocket,
v is the velocity of the rocket, and
c is the speed of light in a vacuum.

By substituting the given values, we can calculate the new length of the rocket.

3. According to the principle of relativity, the force acting on a particle is the same observed by all inertial frames of reference. This means that the laws of physics, including force, are consistent irrespective of the observer's motion.

In the case of two observers in two inertial frames of reference, if they are observing the same particle, the force experienced by the particle will be the same according to both observers. This is because the laws of physics are invariant under Galilean transformations when v<<c (i.e., the relative velocity is much smaller than the speed of light).

4. To find the relative velocity of spaceship B with respect to spaceship A, we need to use the relativistic velocity addition formula. This formula accounts for the fact that velocities do not simply add up linearly in special relativity.

The formula for calculating the relative velocity (v) between two objects moving in opposite directions with velocities u and w, respectively, is given by:

v = (u + w) / (1 + (u * w / c^2))

Where:
v is the relative velocity,
u is the velocity of spaceship A,
w is the velocity of spaceship B, and
c is the speed of light in a vacuum.

By substituting the given values, we can determine the relative velocity of spaceship B with respect to spaceship A.

5. The height of the satellite above the Earth can be calculated using the formula for the period of a satellite's orbit. The period of a circular orbit can be related to the satellite's height (h) and the radius of the Earth (R) as follows:

T = 2π√((R + h)^3 / (g * R^2))

Where:
T is the period of the satellite's orbit,
π is a mathematical constant (approximately 3.14159),
R is the radius of the Earth,
h is the height of the satellite above the Earth, and
g is the acceleration due to gravity at the orbit of the satellite.

By rearranging the formula, we can solve for the satellite's height (h) using the given values of T (90 minutes) and g (9.8 m/s^2), as well as the known radius of the Earth.