3.A certain radar installation tracks airplanes by transmitting electromagnetic radiation of wavelength 3 cm.
a) Show that the frequency of this radiation is 10 GHz.
b) Show that the time required for a pulse of radar waves to reach an airplane 5 km away and return is 3.3 x 10-5 s.
Wave equation:
frequency*wavelength=speed of light
Distance equation:
Distance=velocity*time
where distance it distance there+return distance.
a) To find the frequency of the radiation, we can use the formula:
frequency (f) = speed of light (c) / wavelength (λ)
Given that the wavelength is 3 cm, we convert it to meters by dividing by 100:
wavelength (λ) = 3 cm / 100 = 0.03 meters
The speed of light is approximately 3 x 10^8 m/s, so we can substitute these values into the formula:
frequency (f) = (3 x 10^8 m/s) / (0.03 meters)
Simplifying this equation, we get:
frequency (f) = 10^10 Hz
Since 1 GHz = 10^9 Hz, we can express the frequency in GHz:
frequency (f) = 10^10 Hz = 10 GHz
Therefore, the frequency of the radiation is 10 GHz.
b) To find the time required for the pulse to reach the airplane 5 km away and return, we need to determine the time it takes for the radar waves to travel one way.
We can use the formula:
time (t) = distance (d) / speed (v)
Given that the distance is 5 km, we convert it to meters by multiplying by 1000:
distance (d) = 5 km x 1000 = 5000 meters
The speed of light is approximately 3 x 10^8 m/s, so we substitute these values into the formula:
time (t) = 5000 meters / (3 x 10^8 m/s)
Simplifying this equation, we get:
time (t) = 1.67 x 10^-5 seconds
Since the pulse needs to travel back and forth, we double this time:
time (t) = 1.67 x 10^-5 seconds x 2 = 3.34 x 10^-5 seconds
Therefore, the time required for a pulse of radar waves to reach an airplane 5 km away and return is approximately 3.3 x 10^-5 seconds.
a) The speed of light in a vacuum is approximately 3 x 10^8 m/s. We can use the formula v = λf, where v is the speed of light, λ is the wavelength, and f is the frequency, to find the frequency.
First, we need to convert the wavelength from centimeters to meters:
Wavelength = 3 cm = 3 x 10^-2 m
Now we can rearrange the formula to solve for f:
f = v / λ
Substituting in the values:
f = (3 x 10^8 m/s) / (3 x 10^-2 m)
f = 10^10 Hz = 10 GHz
Therefore, the frequency of the radiation is 10 GHz.
b) To find the time required for a pulse of radar waves to travel to the airplane and back, we can use the formula:
Time = 2 x Distance / Speed
The distance from the radar installation to the aircraft is given as 5 km, which is equal to 5 x 10^3 m. The speed of light is approximately 3 x 10^8 m/s.
Substituting the values into the formula:
Time = 2 x (5 x 10^3 m) / (3 x 10^8 m/s)
Time = 1 x 10^-5 s = 10^-5 s
Therefore, the time required for the radar waves to reach the airplane 5 km away and return is 3.3 x 10^-5 s.
a) To find the frequency of electromagnetic radiation, we can use the equation:
speed of light = wavelength * frequency.
The speed of light is approximately 3 x 10^8 meters per second (m/s). However, we need to convert the wavelength from cm to meters.
Given that the wavelength is 3 cm, we can convert it to meters by dividing by 100:
wavelength = 3 cm ÷ 100 = 0.03 meters.
Now, we can rearrange the equation to solve for frequency:
frequency = speed of light ÷ wavelength.
frequency = 3 x 10^8 m/s ÷ 0.03 m = 1 x 10^10 Hz.
However, since the question asks for the frequency in gigahertz (GHz), we need to convert the frequency to GHz:
frequency = 1 x 10^10 Hz ÷ 10^9 = 10 GHz.
Therefore, the frequency of this radiation is 10 GHz.
b) To find the time required for a pulse of radar waves to reach an airplane 5 km away and return, we can use the equation:
time = distance ÷ speed.
The distance is given as 5 km, so we need to convert it to meters:
distance = 5 km * 1000 m/km = 5000 meters.
The speed of radar waves is the same as the speed of light, which is approximately 3 x 10^8 meters per second (m/s).
Therefore, we can calculate the time:
time = 5000 meters ÷ (3 x 10^8 m/s) = 1.67 x 10^(-5) seconds.
However, since the question asks for the time in scientific notation, we can rewrite the answer as:
time = 1.67 x 10^(-5) s.
Therefore, the time required for a pulse of radar waves to reach an airplane 5 km away and return is 3.3 x 10^(-5) s.