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Question 13 of 36

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SAT Subject Test

Which of the graphs best represents the kinetic energy K of an elementary particle as a function of its speed v, where c is the speed of light?

Option A presents a figure of a graph in the first quadrant. The origin is labeled O, the horizontal axis is labeled v, and the vertical axis is labeled K. A dashed vertical line begins on the horizontal axis two thirds of the way to the right, and extends to the top of the graph. The label c is on the horizontal axis, at the location of the vertical dashed line. A solid concave up curve begins about one quarter of the way up the vertical axis. The curve extends up and to the right. The line starts with a shallow slope and gradually becomes steeper and approaches a vertical line at the dashed vertical line.
Option B presents a figure of a graph in the first quadrant. The origin is labeled O, the horizontal axis is labeled v, and the vertical axis is labeled K. A dashed vertical line begins on the horizontal axis two thirds of the way to the right, and extends to the top of the graph. The label c is on the horizontal axis, at the location of the vertical dashed line. A solid concave up curve begins at the origin. The curve extends up and to the right. The curve starts with a shallow slope and gradually becomes steeper and approaches a vertical line at the dashed vertical line.
Option C presents a figure of a graph in the first quadrant. The origin is labeled O, the horizontal axis is labeled v, and the vertical axis is labeled K. A dashed vertical line begins on the horizontal axis two thirds of the way to the right, and extends to the top of the graph. The label c is on the horizontal axis, at the location of the vertical dashed line. A solid concave up curve begins about one quarter of the way up the vertical axis. The line extends up and to the right. The line starts at about a 45 degree angle and gradually becomes steeper and approaches a vertical line at the dashed vertical line.
Option D presents a figure of a graph in the first quadrant. The origin is labeled O, the horizontal axis is labeled v, and the vertical axis is labeled K. A dashed vertical line begins on the horizontal axis two thirds of the way to the right, and extends to the top of the graph. The label c is on the horizontal axis, at the location of the vertical dashed line. A solid straight line begins at the origin, extends up and to the right, and crosses the dashed line approximately three quarters of the way up the vertical axis.
Option E presents a figure of a graph in the first quadrant. The origin is labeled O, the horizontal axis is labeled v, and the vertical axis is labeled K. A dashed vertical line begins on the horizontal axis two thirds of the way to the right, and extends to the top of the graph. The label c is on the horizontal axis, at the location of the vertical dashed line. A solid concave up curve begins at the origin. The line extends up and to the right. The line starts with a shallow slope and gradually becomes steeper. It crosses the dashed line about one third of the way up the vertical axis and continues to get steeper.
 

Select an Answer

graph (A)

Correct Answer: 
No

graph (B)

Correct Answer: 
Yes

graph (C)

Correct Answer: 
No

graph (D)

Correct Answer: 
No

graph (E)

Correct Answer: 
No

The correct answer is graph (B). In classical mechanics, which is appropriate to apply at low speeds, the kinetic energy K of a particle is given as a function of its speed v by K equals left parenthesis 1 half right parenthesis m v squared, where m is the mass of the particle. So K must be zero when v is zero. But according to special relativity, which must be applied when speeds are near the speed of light c, the speed of a particle can approach but never reach or exceed c. K continues to increase as the speed increases, but the curve can only get closer to the line at v equals c and cannot intersect it. Graph (B) is the only one that satisfies both criteria; that is, it is zero when v  is zero and it approaches v equals c asymptotically.

Question Difficulty: 
easy