Suppose you measure the velocity of a mosquito to be 1.3 m/s with an error of .2 m/s. We approximate the mosquito to weight approximately equal to a grain of rice, which is on the order of magnitude of 10−6 kg. What is the error in the kinetic energy? Assume there is no rotational velocity. In the second answer box, enter units involving m, s, and kg to denote meters, seconds, and kilograms respectively. (Do not use J for Joules or N for Newtons.)
Answer with 2 significant figures of accuracy, ( i.e.with 2 nonzero numbers.)
Hmm... the error in the kinetic energy of a mosquito... well, let's see if I can whip up a joke to help with this one!
Why did the mosquito go to acting school?
Because it wanted to become a bloodsucking star!
Now, let's calculate the error in kinetic energy. The kinetic energy of an object is given by the formula KE = (1/2)mv^2, where m is the mass and v is the velocity. We have the velocity of the mosquito as 1.3 m/s with an error of 0.2 m/s, and the mass is approximately 10^-6 kg.
To find the error in kinetic energy, we need to determine the maximum and minimum possible values of kinetic energy. We'll use the maximum velocity (1.3 + 0.2 = 1.5 m/s) and the minimum velocity (1.3 - 0.2 = 1.1 m/s) to calculate these values.
The maximum kinetic energy would be KE_max = (1/2) * (10^-6 kg) * (1.5 m/s)^2, and the minimum kinetic energy would be KE_min = (1/2) * (10^-6 kg) * (1.1 m/s)^2.
Now, let's plug in the numbers and calculate the values!
KE_max = (1/2) * (10^-6 kg) * (1.5 m/s)^2
KE_max ≈ 1.125 x 10^-12 kg * m^2 / s^2
KE_min = (1/2) * (10^-6 kg) * (1.1 m/s)^2
KE_min ≈ 6.05 x 10^-13 kg * m^2 / s^2
The error in kinetic energy would be the maximum value minus the minimum value:
Error in KE = KE_max - KE_min
Error in KE ≈ (1.125 x 10^-12 kg * m^2 / s^2) - (6.05 x 10^-13 kg * m^2 / s^2)
Error in KE ≈ 5.20 x 10^-13 kg * m^2 / s^2
So, the error in kinetic energy is approximately 5.20 x 10^-13 kg * m^2 / s^2.
And that, my friend, is the buzzing truth!
The kinetic energy (KE) of an object is given by the formula:
KE = (1/2)mv^2
where m is the mass and v is the velocity of the object.
Given that the velocity of the mosquito is 1.3 m/s with an error of 0.2 m/s, we can calculate the maximum and minimum velocities:
Maximum velocity: 1.3 + 0.2 = 1.5 m/s
Minimum velocity: 1.3 - 0.2 = 1.1 m/s
The mass of the mosquito is approximately 10^-6 kg.
Calculating the kinetic energy for the maximum velocity:
KE_max = (1/2)(10^-6 kg)(1.5 m/s)^2 = 1.125 x 10^-6 J
Calculating the kinetic energy for the minimum velocity:
KE_min = (1/2)(10^-6 kg)(1.1 m/s)^2 = 6.05 x 10^-7 J
The error in kinetic energy can be determined by finding the difference between the maximum and minimum values:
Error in KE = KE_max - KE_min = 1.125 x 10^-6 J - 6.05 x 10^-7 J = 5.1975 x 10^-7 J
Rounding the error to 2 significant figures gives:
Error in KE = 5.2 x 10^-7 J
Therefore, the error in the kinetic energy is 5.2 x 10^-7 J (Joules).
To find the error in the kinetic energy, we can use the formula for kinetic energy:
KE = (1/2) * m * v^2
where KE is the kinetic energy, m is the mass, and v is the velocity. Given that the velocity of the mosquito is 1.3 m/s with an error of 0.2 m/s, and the mass is approximately 10^-6 kg, we can calculate the error in the kinetic energy as follows:
1. Calculate the kinetic energy using the maximum velocity:
KE_max = (1/2) * m * (v + Δv)^2
where Δv is the error in velocity. Substituting the given values:
KE_max = (1/2) * (10^-6 kg) * (1.3 m/s + 0.2 m/s)^2
= 1/2 * (10^-6) kg * (1.5 m/s)^2
= 1/2 * (10^-6) kg * 2.25 m^2/s^2
= 1.125 * 10^-6 kg * m^2/s^2
2. Calculate the kinetic energy using the minimum velocity:
KE_min = (1/2) * m * (v - Δv)^2
Substituting the given values:
KE_min = (1/2) * (10^-6 kg) * (1.3 m/s - 0.2 m/s)^2
= 1/2 * (10^-6) kg * (1.1 m/s)^2
= 1/2 * (10^-6) kg * 1.21 m^2/s^2
= 0.605 * 10^-6 kg * m^2/s^2
3. Calculate the error in kinetic energy:
ΔKE = KE_max - KE_min
Substituting the calculated values:
ΔKE = (1.125 * 10^-6 kg * m^2/s^2) - (0.605 * 10^-6 kg * m^2/s^2)
= 0.52 * 10^-6 kg * m^2/s^2
Therefore, the error in the kinetic energy is 0.52 * 10^-6 kg * m^2/s^2, which can be written as 5.2 x 10^-7 kg * m^2/s^2 using 2 significant figures.