1. A rocket was found to be 150m when measured on earth.It then leaves and move at a constant velocity of 2 x 10m/s. What is its new length?
2. Two spaceships A and B are moving in opposite directions, A with a speed of 0.9c while B with a speed of 6c. Find the relative velocity of B with respect to A.
3.Show that the force acting on a particle as observed by two observers in the inertial frames if reference is the same is the same. Given v<<c
1. To find the new length of the rocket, we can use the concept of length contraction in special relativity. According to the theory of relativity, as an object moves faster, its length in the direction of motion appears shorter when observed from a stationary reference frame.
The formula for length contraction is given by:
L' = L * sqrt(1 - (v^2 / c^2))
L' represents the contracted length, L represents the original length, v represents the velocity of the rocket, and c represents the speed of light.
In this case, the original length of the rocket measured on Earth is L = 150m, and the velocity of the rocket is v = 2 x 10 m/s (constant velocity). The speed of light is approximately c = 3 x 10^8 m/s.
Plugging in these values into the length contraction formula:
L' = 150m * sqrt(1 - ((2 x 10 m/s)^2 / (3 x 10^8 m/s)^2))
Simplifying the equation gives us the new length:
L' ≈ 150m * sqrt(1 - 4 x 10^-16)
The value inside the square root is extremely close to 1, since the velocity of the rocket is much smaller than the speed of light (v << c). Therefore, we can approximate length contraction to:
L' ≈ 150m * sqrt(1 - 0)
L' ≈ 150m
So the new length of the rocket, when observed from a reference frame moving at a constant velocity of 2 x 10 m/s, would be approximately 150m.
2. To find the relative velocity of spaceship B with respect to spaceship A, we need to use the velocity addition formula in special relativity. In relativity, velocities cannot simply be added linearly; instead, we use a relativistic addition formula to account for the effects of time dilation and length contraction.
The velocity addition formula is given by:
v_rel = (v1 + v2) / (1 + (v1*v2) / c^2)
In this case, spaceship A is moving with a speed of 0.9c (where c is the speed of light) and spaceship B is moving with a speed of 6c. We want to find the relative velocity of B with respect to A, so we plug these values into the formula:
v_rel = (0.9c + 6c) / (1 + (0.9c * 6c) / c^2)
Simplifying the equation gives us the relative velocity:
v_rel = (0.9c + 6c) / (1 + 0.9*6)
v_rel = 6.9c / 6.9
v_rel = c
Therefore, the relative velocity of spaceship B with respect to spaceship A is equal to the speed of light, c.
3. According to the principle of relativity, the laws of physics remain the same in all inertial reference frames. This means that if we have two observers in different inertial frames, the forces acting on a particle will be the same, as long as they are in equilibrium and the same set of physical laws are applied to describe the system.
Given that v<<c, which means the velocity of the particle is much smaller compared to the speed of light, we can consider the system from classical mechanics, where the laws of motion provided by Newton's laws are valid. In this case, the force acting on an object is given by Newton's second law:
F = m * a
Where F is the force, m is the mass of the particle, and a is the acceleration of the particle as observed by each observer.
Since both observers are in inertial frames, the accelerations observed by each observer will be the same, as long as they are measuring the accelerations with respect to their own reference frames. Therefore, the forces acting on the particle, as observed by two different observers in the same inertial frames of reference, will be the same.