You're investigating a crash in which a 1640 -kg Nissan Leaf...

You're investigating a crash in which a 1640 -kg Nissan Leaf electric car and a 3220 -kg Toyota Land Cruiser SUV collided at right angles in an intersection. The combined wreckage skidded $17.6 \mathrm{~m}$ before stopping. You measure the coefficient of friction between tires and road and find it to be $0.697 .$ Show that at least one car must have exceeded the $70-\mathrm{km} / \mathrm{h}$ speed limit at the intersection. You'll need to consider each car separately, assuming that it was at the speed limit and finding the other car's speed, which you should report in your answer.

Key Concepts

Work-Energy Principle

This principle states that the work done on an object is equal to its change in kinetic energy. In collision analysis, it is used to relate the distance over which friction acts (which does negative work) to the reduction in kinetic energy of the vehicles. By equating the work done by friction to the loss of kinetic energy, one can derive relationships between speed, friction coefficient, and stopping distance.

Friction and Deceleration

Friction, characterized by the coefficient of friction, is the force that opposes motion, causing objects to decelerate. In vehicle dynamics, the friction force is proportional to the normal force and, thus, to the weight of the vehicle. This force determines the rate of deceleration, which is critical when analyzing stopping distances and the speeds required to achieve a given skid length.

Inelastic Collision Dynamics

In inelastic collisions, objects collide and some of the kinetic energy is dissipated as other forms of energy, such as heat or deformation energy, while momentum is conserved. This concept is pivotal in crash investigations since after the impact, the vehicles often stick together or move as a combined mass, altering the energy distribution and affecting the post-collision skid distance.

Momentum Conservation

Momentum conservation is a fundamental principle stating that the total momentum of a closed system remains constant before and after a collision. In crash analysis, applying momentum conservation helps determine the relationship between the speeds and masses of the involved vehicles at the moment of impact, particularly when analyzing scenarios where kinetic energy is not conserved due to inelastic collision effects.

Transcript

00:01 Here we're going to look at an example of an inelastic collision.

00:05 And we'll have to use a little bit of kinematics or energy conservation as well.

00:13 But the idea is that in an inelastic collision, two objects stick together, and they move off together after they collide.

00:27 So on top of conserving momentum, the collision has the stickiness to it.

00:34 And the situation is this.

00:35 There's two vehicles that collide at right angles at an intersection, and the skid marks on the ground are used as a diagnostic for how fast they were going prior to the collision.

00:50 So the skid marks are created when the cars ' tires no longer spin, and the two cars leave friction marks on the ground.

01:02 So the idea is conservation of momentum says that the total momentum beforehand, so the two masses times the velocities, added together, provide one object at the end with a single velocity, with the two masses stuck together.

01:35 That's exactly what you do in an inelastic collision.

01:39 Here because the two cars were traveling at 90 degrees to each other, we can take the magnitude of both sides, and we have m1 v1 squared plus m2 v2 squared, are like two components of a vector, and we can take a square root if we want, m1 plus m2 vf, without its vector sign on it.

02:25 And then we can solve for the final vf.

02:31 Now, we don't know anything about the initial velocities of either of those two vehicles.

02:40 So we can't solve for v final or do anything really too specific with this equation.

02:50 But there are some dynamic relationships that we do know about.

02:55 Namely that the car came to rest in 17 .6 meters that they came to rest as a hole.

03:08 So we're going to use the work energy theorem, and we know that the kinetic energy, that v2 final squared minus v1 final squared is equal to, the force of friction with a negative sign times the distance slid over the total mass times two.

03:51 So this is changing kinetic energy equal work done by friction.

04:02 And i've moved over the one -half and the m in the kinetic energy.

04:08 So what these two velocities are is v2 final is a zero as the cars came to rest.

04:20 And the v1 final is the speed that the two cars had as they collided right after collision.

04:33 So it's the result of the collision, and it's this v final that we have an expression for up above.

04:44 So we'll kind of simplify all that out...

Please give Ace some feedback