Did you ever wonder what it would be like to try a stunt like Evel Knievel, jumping a motorcycle over the Grand Canyon or a string of cars? Fred found out, but he didn't get to enjoy it very long.
Fred was headed West on a rural paved road late at night after having several drinks too many. Either not seeing or ignoring the stop sign, he went right through the intersection. The gravel road straight ahead was a downward slope, which would have caused control problems even without the gravel surface. Fred lost control of his flatbed pickup, running off the road to the right into a ditch and up an embankment.
He braked about 27 ft prior to hitting the embankment, going up and over and becoming airborne as he left the top. The embankment imparted a slight rotational velocity to the pickup. After traveling 40 ft, the truck was inclined at about 32į when the driver's side door hit a 12-inch-diameter wooden power pole and was knocked off. The pole was broken in two places. The pickup continued airborne for another 30 ft prior to hitting a 5-inch-diameter pecan tree, shearing it off. It was then airborne for another 30 ft prior to hitting the ground on a level 4.2 ft below the embankment, tumbling for another 46 ft prior to coming to rest. Fred had massive injuries and did not survive. We would like to know his speed prior to braking.
This is obviously a complex situation not allowing a detailed solution here, but it is interesting to outline a method that can be used to solve this problem. The airborne free-fall of a vehicle may be analyzed in many cases by considering the vehicle to be a point mass, and using the equations for ballistic trajectories given here:
X = V0 Cos0t and
Y = h + V0 Sin0t - 0.5 g t2
X and Y are the horizontal and vertical coordinates, U is the angle of departure, V0 is the initial velocity, h is the initial vertical position, g is the gravitational constant, and t is the time. In order to solve the trajectory equations, it is usually necessary to estimate one of the variables, usually the initial angle of the velocity vector. Since an estimate of the minimum speed may be obtained by using 45į for the initial angle, this is often done. In this case there are three such trajectories interrupted by impacts with the pole and the tree. This suggests that the problem could be analyzed in segments by working backwards as outlined below:
Calculate the work done by tumbling after ground impact, and set this equal to the kinetic energy after impact to determine the speed after ground impact.
Use the speed after ground impact to determine the velocity prior to ground impact. This will be the velo-city at the end of the third trajectory.
Calculate the energy required to fracture the pecan tree, and use this to determine the velocity just prior to tree impact.
Use the trajectory equations to determine the velocity just after pole impact.
Use the energy required to fracture the pole and crush the pick-up to determine the velocity prior to pole impact.
Use the trajectory equations to determine the velocity when leaving the embankment.
Use the work done by rolling friction and braking and the increase in potential energy when climbing the embankment to determine the initial velocity.
After making several simplifying assumptions, it is determined that the speed prior to braking was about 50 mph, perhaps less than you might have guessed.