Relativistic Addition of Velocities Quiz

PDF page 1350: as the water’s velocity relative to the riverbank. (credit: abkfenris, Flickr) If you’ve ever seen a kayak move down a fast-moving river, you know that remaining in the same place would be hard. The river current pulls the kayak along. Pushing the oars back against the water can move the kayak forward in the water, but that only accounts for part of the velocity. The kayak’s motion is an example of classical addition of velocities. In classical physics, velocities add as vectors. The kayak’s velocity is the vector sum of its velocity relative to the water and the water’s velocity relative to the riverbank.

PDF page 1350: For simplicity, we restrict our consideration of velocity addition to one-dimensional motion. Classically, velocities add like regular numbers in one-dimensional motion. (See Figure 28.14.) Suppose, for example, a girl is riding in a sled at a speed 1.0 m/s relative to an observer. She throws a snowball first forward, then backward at a speed of 1.5 m/s relative to the sled. We denote direction with plus and minus signs in one dimension; in this example, forward is positive. Let be the velocity of the sled relative to the Earth, the velocity of the snowball relative to the Earth-

PDF page 1350: For simplicity, we restrict our consideration of velocity addition to one-dimensional motion. Classically, velocities add like regular numbers in one-dimensional motion. (See Figure 28.14.) Suppose, for example, a girl is riding in a sled at a speed 1.0 m/s relative to an observer. She throws a snowball first forward, then backward at a speed of 1.5 m/s relative to the sled. We denote direction with plus and minus signs in one dimension; in this example, forward is positive. Let be the velocity of the sled relative to the Earth, the velocity of the snowball relative to the Earth-

PDF page 1351: The second postulate of relativity (verified by extensive experimental observation) says that classical velocity addition does not apply to light. Imagine a car traveling at night along a straight road, as in Figure 28.15. If classical velocity addition applied to light, then the light from the car’s headlights would approach the observer on the sidewalk at a speed . But we know that light will move away from the car at speed relative to the driver of the car, and light will move towards the observer on the sidewalk at speed , too.

PDF page 1351: The second postulate of relativity (verified by extensive experimental observation) says that classical velocity addition does not apply to light. Imagine a car traveling at night along a straight road, as in Figure 28.15. If classical velocity addition applied to light, then the light from the car’s headlights would approach the observer on the sidewalk at a speed . But we know that light will move away from the car at speed relative to the driver of the car, and light will move towards the observer on the sidewalk at speed , too.

PDF page 1352: Either light is an exception, or the classical velocity addition formula only works at low velocities. The latter is the case. The correct formula for one-dimensional relativistic velocity addition is

PDF page 1352: where is the relative velocity between two observers, is the velocity of an object relative to one observer, and is the velocity relative to the other observer. (For ease of visualization, we often choose to measure in our reference frame, while someone moving at relative to us measures .) Note that the term becomes very small at low velocities, and gives a result very close to classical velocity addition. As before, we see

PDF page 1352: that classical velocity addition is an excellent approximation to the correct relativistic formula for small velocities. No wonder that it seems correct in our experience.

PDF page 1353: Strategy Because the light and the spaceship are moving at relativistic speeds, we cannot use simple velocity addition. Instead, we can determine the speed at which the light approaches the Earth using relativistic velocity addition. Solution

PDF page 1353: Velocities cannot add to greater than the speed of light, provided that is less than and does not exceed . The following example illustrates that relativistic velocity addition is not as symmetric as classical velocity addition.

PDF page 1353: Velocities cannot add to greater than the speed of light, provided that is less than and does not exceed . The following example illustrates that relativistic velocity addition is not as symmetric as classical velocity addition.

PDF page 1354: Discussion The minus sign indicates velocity away from the Earth (in the opposite direction from ), which means the canister is heading towards the Earth in part (a) and away in part (b), as expected. But relativistic velocities do not add as simply as they do classically. In part (a), the canister does approach the Earth faster, but not at the simple sum of . The total velocity is less than you would get classically. And in part (b), the canister moves away from the Earth at a velocity of , which is faster than the you would expect classically. The velocities are not even symmetric. In part (a) the canister moves faster than the ship relative to the Earth, whereas in part (b) it moves slower than the ship.

PDF page 1354: Discussion The minus sign indicates velocity away from the Earth (in the opposite direction from ), which means the canister is heading towards the Earth in part (a) and away in part (b), as expected. But relativistic velocities do not add as simply as they do classically. In part (a), the canister does approach the Earth faster, but not at the simple sum of . The total velocity is less than you would get classically. And in part (b), the canister moves away from the Earth at a velocity of , which is faster than the you would expect classically. The velocities are not even symmetric. In part (a) the canister moves faster than the ship relative to the Earth, whereas in part (b) it moves slower than the ship.