HSC Physics students have difficulty with length contraction problems in the Space section of the course. A graded set of tutorial questions is given below to assist students in length contraction problems. The questions are numerical, so that students can gain an understanding of these concepts by attaching numbers to them.
Length. These questions compare the lengths Lo (the proper length or rest length of the object, being the length that is measured when the the observer is at rest relative to the object) and Lv (the length measured by an observer moving at a constant velocity v relative to the object).
Lv = Lo√(1 - v2/c2)
It is important to note that L is the length component in the direction of the relative velocity v. The relative motion causes a contraction in length Lv.
- Fermilab's Tevatron particle accelerator is 6.3 km long. A proton moves at a constant speed of 2.3x108 m/s through the accelerator. Determine the length of the accelerator in the reference frame of the moving proton.
- An electron moving at 2.9x108 m/s travels between two electrodes that are 6.5 cm apart in the reference frame of the laboratory. Find the distance between the electrodes in the reference frame of the moving electron.
- Relative to an observer in the laboratory a metre stick has a velocity component of 0.97c parallel to its length. Determine the length of the stick according to the laboratory reference frame.
- A straight rod lies along the x axis has a rest length of 35 cm. The rod moves along the x axis at a speed of 2.8x108 m/s relative to the labortory reference frame. Determine the length of the rod in the laboratory frame of reference.
- The length of a spaceship is contracted by 75% when measured from a reference frame. What is the speed of the spaceship relative to the reference frame?
- Muons approach the Earth at a speed of 0.95c. The muon travels a distance of 200 km in its own reference frame. What distance does the muon travel in the refernce frame of the Earth?
- A rocket X of rest length 80 m moves in a straight line at a speed of 0.92c relative to a rocket Y of rest length 60 m moving in the opposite direction. What are the length of Y according to X and the length of X according to Y?
- A star is 6.5 light years from the Earth. A spaceship travels to this star at a constant speed of 0.93c. Determine the distance travelled by the spaceship according to its own reference frame.
- A metre stick at rest in a reference frame S' makes an angle of 40° with the x' axis. A person in a different reference frame S determines this angle to be 60°. Determine the speed of S' relative to S.
- A spaceship is moving away from the Earth at a speed of 0.87c. When the ship is at a distance of 3.6x108 km from the Earth as measured in the Earth's reference frame a radio signal is sent to the spaceship from the Earth. Give the location of the spaceship in the Earth's reference frame when the signal is received.