Question

1) Find the uncertainty in kinetic energy. Kinetic energy depends on mass and velocity according to this function E(m,v) = 1/2 m v². Your measured mass and velocity have the following uncertainties     2.58 kg and   0.36 m/s. What is is the uncertainty in energy,  , if the measured mass, m = 4.75 kg and the measured velocity, v = -3.76 m/s? Units are not needed in your answer

2)Find the uncertainty in kinetic energy. Kinetic energy depends on mass and velocity according to this function E(m,v) = 1/2 m v². Your measured mass and velocity have the following uncertainties     0.47 kg and   2.48 m/s. What is is the uncertainty in energy,  , if the measured mass, m = 3.95 kg and the measured velocity, v = -22.69 m/s? Units are not needed in your answer.

3) The propagation of uncertainty formula for the equation y = ax^2 is

where ,and  . and . The values  and  are the uncertainties on a and x respectively.

If a = -4.5 +/- 0.4 and x = -4.7+/-0.7 then what is the uncertainty on y

4) After a million measurements of thing x, we find a sample mean of 60.45 and standard deviation of 3.24. What chance, in percent (0-100) does the next measurement have of being outside 3 standard deviations from the mean? Do not include the percent sign.

5) Find the uncertainty in a calculated average speed from the measurements of distance and time. Average speed depends on distance and time according to this function v(t,x) = x/t. Your measured distance and time have the following values and uncertainties x = 5.1 meters,   2.8 meters and t = 9.1 seconds and  0.2 seconds. What is the uncertainty in the average speed,  ? Units are not needed in your answer.

Answer 1

1)

We have the Kinetic energy,

So,the relative error in kinetic energy,

We have given that,uncertainity in mass,

Uncertainity in velocity,

measured mass,

measured velocity,

The value of kinetic energy,

so,From the equation,

implies,

So,uncertainity in kinetic energy,

ie,

2)

We have the Kinetic energy,

So,the relative error in kinetic energy,

We have given that,uncertainity in mass,

Uncertainity in velocity,

measured mass,

measured velocity,

The value of kinetic energy,

so,From the equation,

implies,

So,uncertainity in kinetic energy,

ie,

Here since the errors are always +ve,uncertainity on kinetic energy,

3)

Here we have the equation,

Also,

So, Also,

And,

Therefore, Also,

We have the relative uncertainity of y,

We have the value of y,

So,the relative error,

So,the uncertainity in y,

5)

Given that the distance,

Uncertainity in distance,

Time taken,

uncertainity in time,

We have the average velocity

So,relative error in velocity,

We have the value of average velocity,

So,eqn becomes,

So,uncertainity in the average velocity,

or,