The percentage errors in the measurement of mass and speed are 2% and 4%respectively. Then find maximum error in the measurement of K.E?
KE = (1/2) m v^2
dKE= (v^2/2)dm + mvdv
dm = .02 m
dv = .04 v
dKE = (v^2/2) .02 m + m v .04 v
= m v^2/2 (.02) + mv^2/2 (.08 )
= 0.10 KE
= 10%
because velocity is squared, it alone counts for 2*4 = 8%
To find the maximum error in the measurement of kinetic energy (K.E), we need to consider the relationships between mass, speed, and K.E.
The formula for kinetic energy is given by:
K.E = (1/2) * m * v^2
Where:
- K.E = Kinetic Energy
- m = Mass
- v = Speed
Now, let's calculate the maximum error in the measurement of K.E.
Given:
Error in the measurement of mass = 2%
Error in the measurement of speed = 4%
To calculate the maximum error, we can analyze the effect of the errors on each quantity individually and then combine them.
1. Effect of error on mass:
The percentage error in the measurement of mass is 2%. This means that the mass can have a maximum error of 2% from its true value. Let's denote this error as Δm.
2. Effect of error on speed:
The percentage error in the measurement of speed is 4%. This means that the speed can have a maximum error of 4% from its true value. Let's denote this error as Δv.
3. Calculation of maximum error in K.E:
To calculate the maximum error in K.E, we'll use the concept of propagation of errors, which states that when two quantities (mass and speed in this case) are multiplied or divided, their relative errors are added.
Relative error in K.E = Relative error in mass + Relative error in speed
Relative error in mass = (error in mass) / (true value of mass) = Δm / m
Relative error in speed = (error in speed) / (true value of speed) = Δv / v
Hence, relative error in K.E = (Δm / m) + (Δv / v)
Now, let's calculate the maximum error in K.E using the given values:
Given:
Relative error in mass = 2% = 0.02
Relative error in speed = 4% = 0.04
Relative error in K.E = (0.02) + (0.04) = 0.06
Maximum error in K.E = (Relative error in K.E) * (true value of K.E)
Note: The true value of K.E is not given in the question, so we cannot determine the exact value of the maximum error in K.E without knowing the true value of K.E. We can only calculate the relative error.
I hope this explanation helps you understand how to find the maximum error in the measurement of kinetic energy.