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 where time is experienced differently by observers in different relative motion or gravitational fields. According to the theory of relativity, when an object is moving relative to an observer, time for that object will appear to move slower relative to a stationary observer.

(b) To show that an interval of time observed in a moving frame of reference is less than the same interval observed in a stationary frame of reference, we can use the concept of time dilation. The equation to calculate time dilation is:

Δt' = Δt / √(1 - v^2/c^2)

where Δt' is the observed time interval in the moving frame, Δt is the observed time interval in the stationary frame, v is the velocity of the moving frame, and c is the speed of light.

By plugging in the appropriate values, you can compare the observed time intervals in the two frames. You will find that the observed time interval in the moving frame will be less than the observed time interval in the stationary frame, indicating time dilation.

2. To find the new length of the rocket moving at a constant velocity of 2 x 10^8 m/s, we need to consider length contraction. According to the theory of relativity, when an object is moving relative to an observer, its length will appear shorter in the direction of motion.

The equation for length contraction is:

L' = L * √(1 - v^2/c^2)

where L' is the observed length in the moving frame, L is the length in the stationary frame, v is the velocity of the moving frame, and c is the speed of light.

By plugging in the given value of L (150 m) and the velocity v (2 x 10^8 m/s), you can calculate the new length of the rocket as observed in the moving frame.

3. According to the principle of relativity, the force acting on a particle is independent of the inertial frame of reference. This means that two observers in different inertial frames will measure the same force acting on a particle.

Given the condition v << c (where v is the relative velocity between the two frames and c is the speed of light), the difference in velocity is small compared to the speed of light. In this case, relativistic effects such as time dilation and length contraction can be neglected, and classical mechanics can be used.

Hence, the force observed by two observers in different inertial frames will be the same.

4. To find the relative velocity of B with respect to A, we can use the relativistic addition of velocities formula. The formula is given by:

v_relative = (v1 + v2) / (1 + (v1 * v2 / c^2))

where v_relative is the relative velocity, v1 and v2 are the velocities of spaceships A and B respectively, and c is the speed of light.

By plugging in the given values for v1 (0.9c) and v2 (0.6c), you can calculate the relative velocity of B with respect to A.

5. To calculate the height of the satellite above the Earth, we can use the formula for orbital period of a satellite. The formula is given by:

T = 2π * √(r^3 / GM)

where T is the period of the satellite (90 minutes = 5400 seconds), r is the radius of the Earth plus the height of the satellite, G is the gravitational constant, and M is the mass of the Earth.

Rearranging the formula, we can solve for r:

r = ((GM * T^2) / (4π^2))^(1/3) - R

where R is the radius of the Earth.

By plugging in the given values for T (5400 seconds), G, M, and R, you can calculate the height of the satellite above the Earth's surface.