During a crash test, a car moving at \(50 \mathrm{km} / \mathrm{h}\) collides with a rigid barrier and comes to a complete stop in 200 ms. The collision force as a function of time is given by \(F=a t^{4}+b t^{3}+c t^{2}+d t\) where \(\quad a=-8.86 \mathrm{GN} / \mathrm{s}^{4}, b=3.27 \mathrm{GN} / \mathrm{s}^{3}, c=-362 \mathrm{MN} / \mathrm{s}^{2}\) and \(d=12.5 \mathrm{MN} / \mathrm{s}\). Find (a) the total impulse imparted by the collision, (b) the average collisional force, and (c) the car's mass.

Understanding how vehicles behave during collisions is not only important for safety but also a fascinating application of physics principles. In automobile crash tests, various physical quantities such as speed, collision time, and forces involved are meticulously analyzed.

In the context of this exercise, the car's initial velocity and collision time frame are employed to assess the forces at play and the car's response to these forces. Observing the conversion of the car's kinetic energy to other forms through the forces exerted by the barrier gives insights into the car's structural integrity and the effectiveness of its safety mechanisms.

To accurately analyze such a scenario, a force-time function is given, highlighting the varying force over the very short duration of a crash. The polynomial nature of the function indicates that the force increases and decreases in a non-linear manner, thus making the calculation of impulse - the integral of force over time - a crucial step in understanding the dynamics of the collision.

The principle of conservation of linear momentum states that if no external forces act on a system, the total linear momentum remains constant. In the case of a crash test, while the car's momentum changes dramatically, the system's overall momentum is conserved when considering an external force, such as the barrier, to be part of the system.

For the car, the initial momentum is given by the product of its mass and velocity before the collision. After the crash, the car comes to a stop, meaning that its final momentum is zero. The barrier's reaction force implies that the impulse, which equals the change in momentum, is crucial to understanding the event's details. By analyzing the given force function and integrating it over the time of the collision, we can deduce the total impulse which, in turn, corresponds to the change in linear momentum of the car.

Newton's second law forms the cornerstone of our understanding of how forces affect the motion of objects. It's elegantly summarized by the equation \( F = ma \), where \( F \) is the net force applied to an object, \( m \) is the object's mass, and \( a \) is the acceleration it undergoes.

In our crash test scenario, this law allows us to link the forces experienced by the car during the collision to the car's deceleration and by extension, to calculate the car's mass. We know from the exercise that the car's final velocity is 0 m/s, and we have measured the collision time. With the average force calculated, we can use these figures to work backward and find the car's mass. Evaluating this relationship emphasizes how fundamental physics laws apply directly to real-world scenarios like crash testing.

The average collision force is a simplified representation of how the force during an event like a car crash is distributed over time. It is computed by dividing the total impulse, the integrated force over the collision duration, by the time interval during which the collision took place.

In practical terms, the average collision force gives us a value that, if applied evenly over the entire collision time, would result in the same momentum change as the actual varying collision force. This is especially useful in crash tests as it provides a single, comprehensive figure that can be used for designing safety features, like airbags and crumple zones, to protect the occupants under such average forces. Calculating and analyzing the average force helps in evaluating vehicle safety standards, and facilitates comparisons between different vehicles or crash scenarios.