Acceleration in One Dimensional Inelastic Collision

[Calvin Pitts]

The 540 kg car has a magnitude of -32 m/s, and the 1400 kg car has a change of -12.4 m/s. Dividing each number by .054 seconds gives accelerations of -600 m/s and -230 m/s (I hope :) )

 

[haruspex]

The 540 kg car has a magnitude of -32 m/s, and the 1400 kg car has a change of -12.4 m/s. Dividing each number by .054 seconds gives accelerations of -600 m/s and -230 m/s (I hope :) )

That looks about right.

 

[Calvin Pitts]

That looks about right.

Works for me! Thank you so much for your help.

 

[Chestermiller]

CAR COLLISION MODEL WITH CRUMPLING

In this development, the energy dissipating bumper crumpling effect is modeled as a viscous damper, representing the front end of each vehicle. The force of a viscous damper is proportional to the velocity difference between the two ends of the damper. The dampers for the two vehicles are taken as identical so that, by Newton's 3rd law, since the forces on the two dampers are the same, the amount of crushing will be the same. Also, the velocity of the interface between the two dampers is always equal to the average of the two vehicle velocities. For this model, the force balances on the two vehicles are given by:
$$m_h\frac{dv_h}{dt}=\frac{k}{2}(v_l-v_h)\tag{1}$$
$$m_l\frac{dv_l}{dt}=-\frac{k}{2}(v_l-v_h)\tag{2}$$
where the subscript l refers to the light vehicle, h refers to the heavy vehicle, v is velocity in the positive x direction, m is mass, and k is the proportionality constant for each damper. The factor of 2 in the denominators takes into account the fact that the overall velocity difference is equal to twice the velocity difference across each individual damper.

The analytic solution to Eqns. 1 and 2 is given by:
$$v_h=v_{\infty}+\frac{m_l}{(m_h+m_l)}(v_{h0}-v_{l0})e^{-\frac{(m_h+m_l)}{2m_hm_l}kt}\tag{3}$$
$$v_l=v_{\infty}+\frac{m_h}{(m_h+m_l)}(v_{l0}-v_{h0})e^{-\frac{(m_h+m_l)}{2m_hm_l}kt}\tag{4}$$
where $v_{\infty}$ is the final wreck velocity at long times, given by:
$$v_{\infty}=\frac{(m_hv_{h0}+m_lv_{l0})}{(m_h+m_l)}\tag{5}$$with $v_{h0}$ and $v_{l0}$ representing the initial velocities before collision.

From these results, it follows that the relative velocity of the two ends of each of the dampers is given by:
$$\Delta v=\frac{ (v_{10}-v_{20})}{2}e^{-\frac{(m_h+m_l)}{2m_hm_l}kt}\tag{6}$$
The crumple distance of each vehicle is then given by:
$$d=\int_0^{\infty}{(\Delta v)dt}=\frac{m_hm_l}{k(m_h+m_l)}(v_{h0}-v_{l0})\tag{7}$$
From this, it follows that the damper constant k is given by:
$$k=\frac{m_hm_l}{d(m_h+m_l)}(v_{h0}-v_{l0})\tag{8}$$
Substituting Eqn. 8 into Eqns. 3 and 4 for the velocities of the two vehicles during the collision yields:
$$v_h=v_{\infty}+\frac{m_l}{(m_h+m_l)}(v_{h0}-v_{l0})e^{-\frac{(v_{h0}-v_{l0})t}{2d}}\tag{9}$$
$$v_l=v_{\infty}+\frac{m_h}{(m_h+m_l)}(v_{l0}-v_{h0})e^{-\frac{(v_{h0}-v_{l0})t}{2d}}\tag{10}$$
Taking the derivatives of these velocities with respect to time, we obtain the accelerations of the two vehicles:
$$a_h=-\frac{m_l}{(m_h+m_l)}\frac{(v_{h0}-v_{l0})^2}{2d}e^{-\frac{(v_{h0}-v_{l0})t}{2d}}\tag{11}$$
$$a_l=+\frac{m_h}{(m_h+m_l)}\frac{(v_{h0}-v_{l0})^2}{2d}e^{-\frac{(v_{h0}-v_{l0})t}{2d}}\tag{12}$$
Note that both accelerations start out large in magnitude, and then rapidly decay exponentially with time.