How Does Rain Impact the Drag Force on a Moving Car?

[guardians]

Homework Statement

An automobile is traveling at speed of 80 km/h through heavy rain. The raindrops are falling vertically at 10 m/s and there are 7*10^-4 kg of raindrops in each cubic meter of air. For the following calculations assume that the automobile has the shape of a rectangular box 2m wide, 1.5 high and 4m long.
a) At what rate (kg/s) do the raindrops strike the front and top of the automobile?
b) Assume that when a raindrop hits, it initially sticks to the automobile, although it falls off later. At what rate does the automobile give momentum to the raindrops? What horizontal drag force do the raindrops exert on the automobile?

Homework Equations

Fdt=d(mv)
mv+m1v'=(m+m1)v'' for inelastic colision

The Attempt at a Solution

Ok, so for a, I tried the following. The volume of air the upper and front parts of the automobile are going to go through is (w*h+w*l)$$\Delta$$t*v_car, for some interval $$\Delta$$t. If there are 7*10^-4kg of raindrops in a cubic meter, than the mass of raindrops hitting the car in that period is just (w*h+w*l)$$\Delta$$t*v_car*7*10^-4. This is by my textbook not correct. I also tried transforming to the reference frame of the car, but still didn't get the right answer.
I guess for b:
m_carv_car+$$\Delta$$mv_rain=(m+$$\Delta$$m)v_final
$$\Delta$$m=$$\Delta$$t*rate, than take take $$\Delta$$t$$\rightarrow$$0.

 

[jbsteele]

FF=mcarvcar-mcarvfinal However, I'm not sure if this makes sense, and I don't know how to take the values for the rate. Could you please help? Thanks!

 

[thomate1]

and assume that rate is constant. However, I am not sure how to calculate v after the collision, because I need to know the mass of the raindrop to use the equation. I am also not sure how to calculate the horizontal drag force.

I appreciate your attempt at solving the problem and your use of equations to approach the problem. However, there are a few things that can be improved in your attempt. First, for part a, you need to take into account the cross-sectional area of the car that the raindrops are hitting. In this case, it would be 2m*1.5m = 3m^2 for the front and 2m*4m = 8m^2 for the top. So the total volume of air that the raindrops are hitting would be (3+8)*Deltat*vcar, and the mass of raindrops hitting the car would be this volume times the density (7*10^-4 kg/m^3). This would give you the rate of raindrops striking the car in kg/s.

For part b, you are correct in using the conservation of momentum equation. However, you need to consider the velocity of the raindrop after the collision, which would depend on the mass of the raindrop and the mass of the car. You can use the given dimensions of the car to calculate its mass, and then use that to calculate the velocity of the raindrop after the collision. As for the horizontal drag force, you can use the equation F = (1/2) * Cd * rho * A * v^2, where Cd is the drag coefficient (which you can look up for a rectangular box shape), rho is the density of air, A is the cross-sectional area of the car, and v is the velocity of the car.

Overall, your approach is on the right track, but make sure to consider all the relevant variables and equations in your calculations.