If a cart hits a wall, does the weight of it affect how it moves, when the center of gravity is constant?

I have a model that represents a bicycle (a wood block with wheels), and I'm balancing the center of gravity so it's the same as a real bike. However, when the center of mass is kept constant, does the weight of it affect the effect torque has on it when it hits a wall?

I'm planning to measure the angle the back wheel bounces up.

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No. When you hit the wall, the bicycle rotates around the front axis. The angular momentum L that you create for an arbitrary number of mass particles is $$L=\Sigma_i(r_i \times m_iv_i) .$$ If you split location r=R+r_i and v=V+v_i with R and V being center of mass location and velocity, respectively, and r_i and v_i deviation from it, then it can be shown that L does not change when the center of mass does not change.