| Date | November 2017 | Marks available | 2 | Reference code | 17N.3.HL.TZ0.8 |
| Level | Higher level | Paper | Paper 3 | Time zone | Time zone 0 |
| Command term | Calculate | Question number | 8 | Adapted from | N/A |
Question
The Schwarzschild radius of a black hole is 6.0 x 105 m. A rocket is 7.0 x 108 m from the black hole and has a clock. The proper time interval between the ticks of the clock on the rocket is 1.0 s. These ticks are transmitted to a distant observer in a region free of gravitational fields.
Outline why the clock near the black hole runs slowly compared to a clock close to the distant observer.
Calculate the number of ticks detected in 10 ks by the distant observer.
Markscheme
this is gravitational time dilation
OR
black hole gives rise to a «strong» gravitational field
clocks in stronger field run more slowly
OR
the clock «signal» is subject to gravitational red-shift
the clock is subject to gravitational red shift
OR
the clock has lost gravitational potential energy in moving close to the black hole
[Max 2 Marks]
ALTERNATIVE 1 (10 ks is in observer frame):
Δt' = \(10000\sqrt {1 - \frac{{6.0 \times {{10}^5}}}{{7.0 \times {{10}^8}}}} \)
9995.7 so 9995 «ticks»
Allow 9996
Allow ECF if 10 is used instead of 10000
ALTERNATIVE 2 (10 ks is in rocket frame):
Δt = \(\frac{{10000}}{{\sqrt {1 - \frac{{6.0 \times {{10}^5}}}{{7.0 \times {{10}^8}}}} }}\)
10004 «ticks»
Allow ECF if 10 is used instead of 10000
