Worksheet: The Lorentz Velocity Transformation

In this worksheet, we will practice calculating the velocity of an object in a reference frame that has a relative speed to the frame that the velocity was measured in.

Q1:

Consider two inertial reference frames, 𝑆 and 𝑆. 𝑆 moves at a speed of 0.50𝑐 relative to 𝑆. An object moves at a speed of 0.20𝑐 in 𝑆. The motion of the object is antiparallel to the motion of frame 𝑆 relative to 𝑆. What is the speed of the object in frame 𝑆? Give your answer as a multiple of 𝑐 to 2 significant figures.

  • A0.64𝑐
  • B0.27𝑐
  • C0.40𝑐
  • D0.20𝑐
  • E0.33𝑐

Q2:

A spaceship fires a projectile toward an enemy spaceship. The projectile moves at a speed of 0.10𝑐 toward the enemy spaceship. Neither spaceship is moving relative to each other, but both are moving at a speed of 0.50𝑐 relative to Earth. How fast does the projectile move toward the enemy ship in the reference frame of Earth? The motion of the projectile is parallel to the motion of the spaceships relative to Earth. Give your answer as a multiple of 𝑐 to 2 significant figures.

  • A0.63𝑐
  • B0.57𝑐
  • C0.55𝑐
  • D0.45𝑐
  • E0.40𝑐

Q3:

Consider two inertial reference frames, 𝑆 and 𝑆. 𝑆 moves at a speed of 0.40𝑐 relative to 𝑆. An object moves at a speed of 0.10𝑐 in 𝑆. The motion of the object is parallel to the motion of frame 𝑆 relative to 𝑆. What is the speed of the object in frame 𝑆? Give your answer as a multiple of 𝑐 to 2 significant figures.

  • A0.29𝑐
  • B0.48𝑐
  • C0.67𝑐
  • D0.50𝑐
  • E0.31𝑐

Q4:

A spaceship fires a projectile toward an enemy spaceship. The projectile moves at a speed of 1.4×10 m/s toward the enemy spaceship in the reference frame of Earth. Both spaceships are moving at a speed of 6.5×10 m/s relative to Earth. How fast does the projectile move toward the enemy ship in the reference frame of the spaceship it was fired from? The motion of the projectile is parallel to the motion of the spaceships relative to Earth. Give your answer to 2 significant figures.

  • A1.8×10 m/s
  • B6.2×10 m/s
  • C2.2×10 m/s
  • D8.3×10 m/s
  • E6.8×10 m/s

Q5:

A spaceship that is returning to Earth sends out radio wave transponder signals every few seconds as it approaches the planet. The spaceship is moving toward Earth at a speed of 0.05𝑐. The radio waves move away from the ship at a speed 𝑐 in the reference frame of the ship. At what speed do the radio waves move away from the ship in the reference frame of Earth?

  • A0.05𝑐
  • B0.54𝑐
  • C𝑐
  • D0.50𝑐
  • E0.95𝑐

Q6:

Consider two inertial reference frames: 𝑆 and 𝑆. 𝑆 moves at a speed of 2.3×10 m/s relative to 𝑆. An object moves at a speed of 8.1×10 m/s in frame 𝑆, parallel to the motion of 𝑆 relative to 𝑆. What is the speed of the object in frame 𝑆? Give your answer, in meters per second, to 2 significant figures.

  • A2.5×10 m/s
  • B1.3×10 m/s
  • C1.9×10 m/s
  • D2.9×10 m/s
  • E1.2×10 m/s

Q7:

The figure shows a graph of the Lorentz velocity transformation from frame 𝑆 to three other frames, 𝑅, 𝑅, and 𝑅. Frame 𝑅 is represented by the red line, frame 𝑅 is represented by the blue line, and frame 𝑅 is represented by the green line. Which frame has the greatest magnitude of speed relative to frame 𝑆?

  • A𝑅
  • B𝑅
  • C𝑅

Q8:

The figure shows a graph of the Lorentz velocity transformation, where 𝑣 is the speed of an object as measured in a reference frame 𝑆 and 𝑣 is the speed of that object as measured in a reference frame 𝑆. 𝑆 moves at a speed 𝑢 relative to 𝑆. Both 𝑣 and 𝑣 are shown as multiples of 𝑐.

What is the value of 𝑢?

  • A𝑐
  • B0.4𝑐
  • C0.2𝑐
  • D0.4𝑐
  • E0.2𝑐

If an object is stationary in frame 𝑆, what is its speed in frame 𝑆?

  • A0.2𝑐
  • B0.4𝑐
  • C0.4𝑐
  • D𝑐
  • E0.2𝑐

If an object is stationary in frame 𝑆, what is its speed in frame 𝑆?

  • A0.2𝑐
  • B0.4𝑐
  • C0.4𝑐
  • D0.2𝑐
  • E𝑐

Q9:

The figure shows a graph of both the Lorentz velocity transformation and the Galilean velocity transformation for two reference frames, S and S, where 𝑣 is the speed of an object as measured in frame S and 𝑣 is the speed of that object as measured in frame S. S moves at a speed 𝑢 relative to S. The red line shows the Lorentz velocity transformation, and the blue line shows the Galilean velocity transformation. Both 𝑣 and 𝑣 are shown as multiples of 𝑐.

What is the value of 𝑢?

  • A0.2𝑐
  • B0.6𝑐
  • C𝑐
  • D0𝑐
  • E0.4𝑐

Using the Galilean velocity transformation, what is the value of 𝑣 when 𝑣=0.8𝑐?

  • A0.2𝑐
  • B0.8𝑐
  • C0.2𝑐
  • D0.8𝑐
  • E𝑐

Using the Lorentz velocity transformation, what is the value of 𝑣 when 𝑣=0.8𝑐? Give your answer to 2 significant figures.

  • A0.38𝑐
  • B0.95𝑐
  • C𝑐
  • D0.95𝑐
  • E0.38𝑐

Using the Galilean velocity transformation, what is the value of 𝑣 when 𝑣=1𝑐?

  • A0.6𝑐
  • B0.4𝑐
  • C0.6𝑐
  • D0.8𝑐
  • E0.4𝑐

Using the Lorentz velocity transformation, what is the value of 𝑣 when 𝑣=1𝑐?

  • A0.6𝑐
  • B1𝑐
  • C1𝑐
  • D0.8𝑐
  • E0.6𝑐

Using the Galilean velocity transformation, what is the value of 𝑣 when 𝑣=0.3𝑐?

  • A0.3𝑐
  • B0.9𝑐
  • C0.9𝑐
  • D0.3𝑐
  • E0.6𝑐

Using the Lorentz velocity transformation, what is the value of 𝑣 when 𝑣=0.3𝑐? Give your answer to 3 significant figures.

  • A0.366𝑐
  • B0.366𝑐
  • C0.577𝑐
  • D0.577𝑐
  • E𝑐

The formula 𝑣=1𝑢1𝑢1 can be used to calculate the speed an object must have in either S or S to have the same speed, but in the opposite direction, in the other reference frame. What speed must an object have in S to have the same speed, but in the opposite direction, in S? Give your answer to 3 significant figures.

  • A0.333𝑐
  • B0.577𝑐
  • C0.333𝑐
  • D𝑐
  • E0.577𝑐

Q10:

The figure shows a graph of the Lorentz velocity transformation, where 𝑣 is the speed of an object as measured in a reference frame S and 𝑣 is the speed of that object as measured in a reference frame S. S moves at a speed 𝑢 relative to S. Both 𝑣 and 𝑣 are shown as multiples of 𝑐.

What is the value of 𝑢?

  • A0.5𝑐
  • B0.8𝑐
  • C0.5𝑐
  • D0.8𝑐
  • E𝑐

Using the graph, determine the value of 𝑣 when 𝑣 is 0.8𝑐.

  • A1𝑐
  • B0.8𝑐
  • C0.5𝑐
  • D0.5𝑐
  • E0.8𝑐

Using the graph, determine the value of 𝑣 when 𝑣 is 0.5𝑐.

  • A0.5𝑐
  • B0.8𝑐
  • C0.8𝑐
  • D1𝑐
  • E0.5𝑐

Q11:

The figure shows a graph of the Lorentz velocity transformation, where 𝑣 is the speed of an object as measured in a reference frame S and 𝑣 is the speed of that object as measured in a reference frame S. S moves at a speed 𝑢 relative to S. Both 𝑣 and 𝑣 are shown as multiples of 𝑐.

What is the value of 𝑢?

  • A𝑐
  • B0.2𝑐
  • C𝑐
  • D0.6𝑐
  • E0.4𝑐

What is the value of 𝑣 when 𝑣 is 1𝑐?

  • A0.2𝑐
  • B1𝑐
  • C0.4𝑐
  • D0.6𝑐
  • E1𝑐

What is the value of 𝑣 when 𝑣 is 1𝑐?

  • A0.6𝑐
  • B0.4𝑐
  • C0.2𝑐
  • D1𝑐
  • E1𝑐

If an object is moving at 0.60𝑐 in frame S, what is its speed in frame S? Give your answer to 2 significant figures.

  • A0.60𝑐
  • B𝑐
  • C0.59𝑐
  • D0.40𝑐
  • E0.26𝑐

Which of the following inequalities is true for an object that is moving at 0.2𝑐 in S?

  • A𝑣>0.2𝑐
  • B𝑣<0.4𝑐
  • C𝑣>0.2𝑐
  • D𝑣<0.2𝑐
  • E𝑣>0.0𝑐

Q12:

Consider two inertial reference frames, S and S. S moves at a speed of 6.9×10 km/s relative to S. An object moves at a speed of 9.8×10 km/s in frame S, antiparallel to the motion of S relative to S. What is the speed of the object in frame S? Give your answer as a multiple of 𝑐 to 2 significant figures.

  • A0.49𝑐
  • B0.60𝑐
  • C0.52𝑐
  • D0.10𝑐
  • E0.83𝑐

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