Dune buggies are popular with some young people since they can be launched in the air with very little effort, simply by speeding up a sand hill. Unfortunately, serious accidents can occur, especially when the physics of the situation is ignored.
A buggy speeding up a slope will be launched like any other projectile if the slope comes to a sudden end. In that case, the trajectory of the buggy will be a parabola as shown in the following sketch.
Alternatively, if the sand hill is round on top, then the wheels of the buggy will remain on the ground if the speed of the buggy is low enough. If the speed is high enough then the trajectory followed by the buggy will be similar to that when it is launched off a ramp. If the sand hill falls away faster than that trajectory, then the buggy will become airborne.
Suppose that the radius of the sandhill near the top is R, the mass of the buggy is M and the speed of the buggy is v. In order for the buggy to remain on the sandhill there needs to be a centripetal force Mv2/R acting down on the buggy to ensure that it follows the same curve as the sandhill. The force of gravity, Mg, will be sufficient to do that if v is small, but not if v is large. If the buggy is at rest at the top of the hill then N = Mg where N is the normal reaction force of the hill on the buggy. If the buggy is travelling at speed v on the hill then Mg – N = Mv2/R so N = Mg – Mv2/R. As v increases, N decreases. If v is large enough then N drops to zero and the buggy becomes airborne. N = 0 when v2 = Rg = 9.8R. For example, if R = 5 m then v = 7 m/s. That is only a fraction faster than the average running speed of adult males.
The danger of becoming airborne is shown in the following figure. Once the front wheel lifts off the sand, N acts only on the back wheel and tips the back end up. The buggy then does a somersault and can land on its front end or its roof.
The first two clips show a less dangerous launch since these buggies do not rotate as much as the third buggy, despite the fact that the N force tends to rotate all of them. In order to reduce or avoid rotation after launch it is necessary to accelerate the buggy as it approaches the launch point. Then there is an additional force, F, acting in a direction parallel to the sand surface, as shown in the next diagram. F is the friction force acting on the back wheel and it acts to accelerate the buggy up the hill. If the driver applied the brakes near the top of the hill, then F would act backwards on the wheels. But if the driver accelerates then F acts forwards and it acts to rotate the buggy in the opposite direction to that caused by the N force. If F is large enough then the buggy can be launched without any rotation. A similar effect occurs when doing a “wheelie” on a bicycle, where the front wheel lifts off the ground if there is a large enough forward friction force on the back wheel.
The following 300 fps movie files show a tennis ball and a block of wood making their way up a cardboard “sand hill” in slow motion. At low speed, the ball and the wood block make it over the hill without becoming airborne. In addition to the N force, friction acts on the bottom surface of the block and can also contribute to rotation of the block, even if it makes it over the hill safely. Block 2 below crashes when it gets to the bottom of the hill! The crude 30 cm scale at the front shows that the radius of the hill is about 15 cm so the ball or the block become airborne at a speed of only about 1.2 m/s. That is a neat physics project for anyone who wants to try it. A remote controlled toy buggy would be even more realistic.
The two 300 fps video clips here show a toy vehicle launched off the edge of a table. The low speed vehicle rotates more rapidly since the force on the rear wheels acts for a longer time after the front wheels become airborne. In both cases, there is no change in the horizontal speed of the vehicle when the vehicle leaves the table since there is no horizontal force on the vehicle (the driver didn’t apply the brakes or try to accelerate).
For more information, see: R. Cross, Launch of a vehicle from a ramp, The Physics Teacher, 49, 410-411 (2011).