A cart of mass m = 0.84 kg on a track undergoes a collision with another cart. It's velocity before and after the collision, vi and vf is measured and found to be vi = 1.32±0.04 m/s and vf = 0.30±0.04 m/s. If the collision lasts a time interval Δt = 0.100±0.006 s, calculate the average force Fave = Δp/Δt. how do you calculate the standard deviation of Fave?
To calculate the standard deviation of Fave, we need to use the concept of propagation of uncertainties. The formula for the average force is Fave = Δp/Δt, where Δp is the change in momentum.
First, let's calculate the change in momentum: Δp = m(vf - vi).
Next, we need to calculate the uncertainty in Δp. Since we have uncertainties in mass, initial velocity, final velocity, and time interval, we need to propagate these uncertainties through the formula.
To do this, we use the formula for combining uncertainties when multiplying or dividing variables:
((σx/x)^2 + (σy/y)^2 + ...) * (x/y/...)^2
where σ represents the uncertainty in the variable, and x, y, ... represent the value of the variable itself.
Applying this formula, the uncertainty in Δp, σΔp, can be calculated as follows:
σΔp = ((σm/m)^2 + (σvf/vf)^2 + (σvi/vi)^2)^0.5 * Δp
Substituting the given values, we have:
σΔp = ((0.04/1.32)^2 + (0.04/0.3)^2 + (0.04/1.32)^2)^0.5 * Δp
Now, we can calculate the uncertainty in Fave, σFave, using the formula for dividing uncertainties:
σFave = (σΔp/Δp)^2 * Fave
Substituting the given values, we have:
σFave = (σΔp/Δp)^2 * Fave = ((0.006/(0.100))^2) * Fave
Finally, we have the formula for the standard deviation of Fave. To calculate it, we multiply σFave by 1.96, which corresponds to a 95% confidence level:
Standard deviation of Fave = 1.96 * σFave
Substituting the calculated value of σFave, we get:
Standard deviation of Fave = 1.96 * ((0.006/(0.100))^2) * Fave
Note: The factor 1.96 is used for a 95% confidence level, but other factors can be used for different confidence levels (e.g., 2.58 for a 99% confidence level).