A car of mass M, traveling at speed v collides head-on with a truck of mass 2M & traveling in the opposite direction at the same speed. Both vehicles lock together and come to a stop. Which, if either, has the most momentum upon impact? Which, if either, undergoes the greatest impulse? Explain.
The truck has more m v because mass is bigger.
Impulse is force times time. There are no external forces on this system. Therefore the force of the truck on the car is equal and opposite to that of the car on the truck. The time is the same. Thus equal but opposite impulse.
To determine which vehicle has the most momentum upon impact, we need to calculate the momentum before and after the collision.
Momentum (p) is calculated using the equation:
p = mass (m) * velocity (v)
Let's assume the mass of the car is M and the velocity is v, and the mass of the truck is 2M (twice the mass of the car) and the velocity is also v (same speed but in opposite direction).
Before the collision, the momentum of the car is given by:
p(car) = M * v
Before the collision, the momentum of the truck is given by:
p(truck) = 2M * (-v) (negative sign indicates opposite direction)
The total momentum before the collision is the sum of individual momenta:
p(total) = p(car) + p(truck)
= M * v + 2M * (-v)
= Mv - 2Mv
= -Mv
Since the total momentum is negative (-Mv), it implies that the direction is opposite to the initial velocity (head-on collision).
After the collision, both vehicles lock together and come to a stop. The final velocity of the combined system will be zero. Since momentum is conserved, we can set up an equation:
p(total) = m(total) * v(final)
Since v(final) is zero, the total momentum after the collision should also be zero:
0 = m(total) * 0
From this equation, we can conclude that the total momentum after the collision is zero.
Now let's determine which vehicle undergoes the greatest impulse. Impulse is the change in momentum and it can be calculated using the equation:
Impulse (J) = Δp = p(final) - p(initial)
Since both vehicles come to a stop, the final momentum (p(final)) is zero. The initial momentum (p(initial)) is -Mv. Thus, the impulse can be calculated as:
J = 0 - (-Mv)
= Mv
Therefore, the car undergoes the greatest impulse, which means it experiences a greater change in momentum compared to the truck.
To summarize:
- The car has the most momentum upon impact, indicated by the negative total momentum (-Mv).
- The car undergoes the greatest impulse, indicated by the impulse value of Mv.