How do I calculate the upper limit and lower limits in order to obtain F(max)and F(min)for the problem below. Using the force F = (1/2)CApv^2 on the car due to air friction.
Consider a car with front cross-sectional area of A=(1.77 ± 0.02)m^2 and a drag coefficient of C = (0.38 ± -0.02) moving with a speed of v = (10.0 ± 0.5)m/s. Suppose that the density of air is p = 1.20 kg/m^3 at the sea level.
max use
C = .38+.02 = .40
A = 1.77+.02 = 1.79
rho = 1.20
v = 10+ .5 = 10.5
for min subtract instead of add the uncertainties
To calculate the upper limit and lower limits for F(max) and F(min) using the force equation F = (1/2)CApv^2, we will substitute the given values with their upper and lower limits.
Given values:
A = (1.77 ± 0.02) m^2
C = (0.38 ± 0.02)
p = 1.20 kg/m^3
v = (10.0 ± 0.5) m/s
For F(max):
Upper limit for A = 1.77 + 0.02 = 1.79 m^2
Upper limit for C = 0.38 + 0.02 = 0.40
Upper limit for v = 10.0 + 0.5 = 10.5 m/s
F(max) = (1/2)CApv^2
= (1/2)*(1.79)*(0.40)*(1.20)*(10.5)^2
≈ 39.64 N
Therefore, the upper limit for F(max) is approximately 39.64 N.
For F(min):
Lower limit for A = 1.77 - 0.02 = 1.75 m^2
Lower limit for C = 0.38 - 0.02 = 0.36
Lower limit for v = 10.0 - 0.5 = 9.5 m/s
F(min) = (1/2)CApv^2
= (1/2)*(1.75)*(0.36)*(1.20)*(9.5)^2
≈ 25.96 N
Therefore, the lower limit for F(min) is approximately 25.96 N.
To calculate the upper limit and lower limits for F(max) and F(min) using the given equation F = (1/2)CApv^2, we will use the principle of error propagation. This principle allows us to estimate the maximum and minimum values of a calculated result by considering the uncertainties (errors) associated with the input values.
Step 1: Determine the formula to calculate the error in the function. In this case, we have F = (1/2)CApv^2, so we need to calculate the error (ΔF) in terms of the errors in A, C, p, and v.
Step 2: Calculate the partial derivatives (∂F/∂A, ∂F/∂C, ∂F/∂p and ∂F/∂v) for the given function with respect to each variable.
∂F/∂A = (1/2)Cpv^2, ∂F/∂C = (1/2)Apv^2, ∂F/∂p = (1/2)CAv^2, ∂F/∂v = CApv
Step 3: Calculate the error (ΔF) using the formula:
ΔF = √[(∂F/∂A * ΔA)^2 + (∂F/∂C * ΔC)^2 + (∂F/∂p * Δp)^2 + (∂F/∂v * Δv)^2]
where ΔA, ΔC, Δp, and Δv represent the uncertainties in A, C, p, and v, respectively.
Step 4: Substitute the values into the equation to find the upper limit and lower limit for F:
F(max) = F + ΔF
F(min) = F - ΔF
Let's apply this calculation to the given problem:
Given values:
A = (1.77 ± 0.02) m^2
C = (0.38 ± 0.02)
p = 1.20 kg/m^3
v = (10.0 ± 0.5) m/s
Step 1: Determine the formula to calculate the error in the function. In this case, ΔF = √[(∂F/∂A * ΔA)^2 + (∂F/∂C * ΔC)^2 + (∂F/∂p * Δp)^2 + (∂F/∂v * Δv)^2]
Step 2: Calculate the partial derivatives:
∂F/∂A = (1/2)Cpv^2
∂F/∂C = (1/2)Apv^2
∂F/∂p = (1/2)CAv^2
∂F/∂v = CApv
Step 3: Substitute the partial derivatives and uncertainties into the equation to calculate ΔF:
ΔF = √[((1/2)Cpv^2 * ΔA)^2 + ((1/2)Apv^2 * ΔC)^2 + ((1/2)CAv^2 * Δp)^2 + (CApv * Δv)^2]
Step 4: Calculate the upper and lower limits for F:
F(max) = F + ΔF
F(min) = F - ΔF