I don't have a calculator or math program like Mathematica at hand, so please help me solve this equation for v numerically:
1 = 18tan(v) - 7938/cos2(v)
or
1 = 18sin(v)sec(v)-7938sec2(v)
Thanks a lot in advance!
Background
This problem is based on solving for the velocity of a car that drove off the road in my hometown in Lillestrøm, Norway, killing the two people in the car. The car came out of a tunnel where it passed a speed camera which didn't register the car's speed. The camera has a lower cutoff at 40 km/h and an upper cutoff at 256 km/h so the car must have been driving faster than 256km/h since it would not have had time to accelerate enough if it was going below 40km/h. It is the first time this has happened. The motorway inside the tunnel very cleverly ends in a roundabout just outside the tunnel opening. The car hit the curb of the central circle of the roundabout and was airborne across the fence, down a hill, cutting the tops off some bushes, passed above another fence, by a walking path and was found 15 meters from the edge of a river running at normal angle to the tunnel motorway and flight path. In total the car flew 90 meters and I assume that its center of gravity must have been 5 meters up in the air at the highest point of the flight path in order to pass all those items.
I figured that estimating the initial velocity of the car would be an interesting problem to solve for high school students or higher that are learning about trig functions, and I wanted to check if my solution is valid before presenting the problem to anybody else. Here goes:
Accident info (in Norwegian):
http://www.dagbladet.no/2012/07/26/nyheter/innenriks/bilulykke/relingen/lillestrom/22688668/
Based on the info from the newspaper I assume the following parameters/variables. I choose to solve for the initial airborne velocity of the car at a slight upwards angle after it hit the curb, rather than starting with the horizontal speed along the motorway. That velocity is given by v0motorway = v0airborne tan (5/45) and can easily be calculated after the initial airborne velocity has been determined. Here I disregard the friction when hitting the curb and air resistance, and I use a simple tangent instead of considering the slightly higher angle due to the ballistic flight path.
Analytical solution
I take these as negligible factors for this rough estimate.
I have the following basic relationships:
I substitute parameters and variables and simplify:
Again, thanks in advance for helping out!
1 = 18tan(v) - 7938/cos2(v)
or
1 = 18sin(v)sec(v)-7938sec2(v)
Thanks a lot in advance!
Background
This problem is based on solving for the velocity of a car that drove off the road in my hometown in Lillestrøm, Norway, killing the two people in the car. The car came out of a tunnel where it passed a speed camera which didn't register the car's speed. The camera has a lower cutoff at 40 km/h and an upper cutoff at 256 km/h so the car must have been driving faster than 256km/h since it would not have had time to accelerate enough if it was going below 40km/h. It is the first time this has happened. The motorway inside the tunnel very cleverly ends in a roundabout just outside the tunnel opening. The car hit the curb of the central circle of the roundabout and was airborne across the fence, down a hill, cutting the tops off some bushes, passed above another fence, by a walking path and was found 15 meters from the edge of a river running at normal angle to the tunnel motorway and flight path. In total the car flew 90 meters and I assume that its center of gravity must have been 5 meters up in the air at the highest point of the flight path in order to pass all those items.
I figured that estimating the initial velocity of the car would be an interesting problem to solve for high school students or higher that are learning about trig functions, and I wanted to check if my solution is valid before presenting the problem to anybody else. Here goes:
Accident info (in Norwegian):
http://www.dagbladet.no/2012/07/26/nyheter/innenriks/bilulykke/relingen/lillestrom/22688668/
Based on the info from the newspaper I assume the following parameters/variables. I choose to solve for the initial airborne velocity of the car at a slight upwards angle after it hit the curb, rather than starting with the horizontal speed along the motorway. That velocity is given by v0motorway = v0airborne tan (5/45) and can easily be calculated after the initial airborne velocity has been determined. Here I disregard the friction when hitting the curb and air resistance, and I use a simple tangent instead of considering the slightly higher angle due to the ballistic flight path.
Analytical solution
I take these as negligible factors for this rough estimate.
g = | 9,8 | m/s2 | ||||
v0 = | v | m/s | v > | 256 | km/h | 71,11111m/s |
t = | t | s | ||||
x = | 90 | m | ||||
y = | 5 | m | ||||
y = v0y*t - 1/2*g*t2 |
x = v0x*t |
v0y = sin(v) |
v0x = cos(v) |
I substitute parameters and variables and simplify:
5 = v0yt - 4,9t2 |
and t = 90/v0x |
<=> |
5 = 90v0y/v0x - 4,9(90/v0x)2 |
<=> |
5 = 90sin(v)/cos(v) - 4,9*902/cos2(v) |
<=> 1 = 18tan(v)-7938/cos2(v) |
<=> |
1 = 18sin(v)sec(v)-7938sec2(v) |
Again, thanks in advance for helping out!