On a frictionless track, a moving cart collides with a stationary cart, and the two stick together and move as one.
Adjust conditions:
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1 kgのAが6 m/sで、静止した2 kgのBに衝突。連結後の速さは？
接触開始からAを左向きに30 Nで0.10秒間押す。押し終えた後の速さは？
The equations explain the phenomenon
1We write the equation of motion for A.mAdvAdt=FA……
2We also write the equation of motion for B.mBdvBdt=FB……
3We apply Newton's third law.FA+FB=0……
4We form an equation for the motion of the 2 carts as a system.mAdvAdt+mBdvBdt=0……
5Since the masses are constant, we rewrite the expression in terms of derivatives of momentum.d(mA vA)dt+d(mB vB)dt=0……
6We combine the sum of the derivatives into the derivative of the total momentum.d(mA vA + mB vB)dt=0……
7A derivative of 0 means that the total momentum is constant.mA vA+mB vB=Constant……
8We apply the result to carts that stick together.mA vA0+mB vB0=(mA + mB) V……
Further study covers external forces, impulse, and energy.
9We consider what happens when an external force acts.Δ(mA vA + mB vB)=∫Fext dt……
10We consider A alone to relate impulse to the change in momentum.mA V − mA vA0=∫FA dt……
11We compare the kinetic energy before and after the collision.½ mA vA0²+½ mB vB0²＞½ (mA + mB) V²……
What this chapter has shown
By Newton's third law, the internal forces between A and B sum to zero. If the net external force on A and B is also zero, their total momentum remains unchanged throughout the collision.
2 m/sThe speed under the initial conditions
When the external force is applied, its impulse of −3 N·s reduces the total momentum from 6 to 3 kg·m/s.
In everyday lifeA rocket moves forward in reaction to pushing gas backward.
Even when the velocities of individual objects change in a collision, their total momentum remains constant if their masses remain unchanged, Newton's third law holds, and no external force acts. This conclusion follows by adding the equations of motion for the individual objects. The magnitudes and detailed forms of the forces the objects exert on each other are not needed, because these forces cancel when the equations are added. If an external force acts, the total momentum changes by an amount equal to its impulse. Kinetic energy decreases in a collision in which the objects stick together, but momentum is conserved independently of this decrease.