The Riveting Physics of Bumper Cars

• If you've ever rode Krachenwagen at Canada's Wonderland, you know the thrill of riding a bumper car, and colliding with others at relatively high speeds
• This classic amusement park ride follows Newton's third law of motion, that states when two objects interact with each other, there is a simultaneous reaction force in the opposite direction that of equal magnitude, resulting in collisions.

Head-on Elastic Collisions

• The rubber bumper that surrounds each car allows for head-on elastic collisions to occur, meaning the both the kinetic energy and momentum from each car is conserved in one dimension
• Electrical energy is supplied to each car from the electrical grid, in which the cars are always touching through a pole on the back of the car
• This electrical energy is partially converted into kinetic energy, which powers each car, and some of it is lost as thermal energy, in the form of sparks. This energy conversion follows the law of conservation, as denoted below: $E_{Ti} = E_{Tf}$

This means that we can use the conservation of momentum equation, as well as the conservation of kinetic energy to solve for the final velocities of the two cars $P = m*v$

$m_1\overrightarrow v_{i1} + m_2 \overrightarrow v_{i2} = m_1\overrightarrow v_{f1} + m_2 \overrightarrow v_{f2}$

$\frac{1}{2}m_1v_{i_1}^2 + \frac{1}{2}m_2v_{i_2}^2 = \frac{1}{2}m_1v_{f_1}^2 + \frac{1}{2}m_2v_{f_2}^2$

• However, since we have two equations with two unknowns, we can use substitution to rearrange each to solve for final velocity, which looks something like this:

  $V_{1F} = V_{1i}(\frac{m_1-m_2}{m_1+m_2}) + V_{2i}(\frac{2m_2}{m_1 + m_2})$

  $V_{2F} = V_{2i}(\frac{m_2-m_1}{m_1+m_2}) + V_{1i}(\frac{2m_1}{m_1 + m_2})$

• The two cars have the same mass, however, the difference in the masses of the drivers results in the lighter driver having a higher final velocity than the heavier one

Impulse

Similarly, we can also take a look at impulse, which is the change in linear momentum, the result of the application of a force over a given time interval

• We can rewrite the momentum equation for impulse using Newton's second law of motion:

    $P = mv$
  

  $\triangle P = m \triangle v$

  $\triangle P=ma \triangle t$

  $\triangle P=F \triangle t$

• Impulse allows drivers to safely bump into each other at pretty high speeds, since the elastic bumpers allow for the collision to be spread out over a longer period of time, reducing the force felt by each driver

• Since impulse is directly proportional to the force and time, the larger the time interval, the smaller the force applied

Impulse Graph