Ball B, moving in the positive direction of an x axis at...

Ball $B$, moving in the positive direction of an $x$ axis at speed $v$, collides with stationary ball $A$ at the origin. $A$ and $B$ have different masses. After the collision, $B$ moves in the negative direction of the $y$ axis at speed $v / 2 .$ (a) In what direction does $A$ move? (b) Show that the speed of $A$ cannot be determined from the given information.

Ball $B$, moving in the positive direction of an $x$ axis at speed $v$, collides with stationary ball $A$ at the origin. $A$ and $B$ have different masses. After the collision, $B$ moves in the negative direction of the $y$ axis at speed $v / 2 .$ (a) In what direction does $A$ move?
(b) Show that the speed of $A$ cannot be determined from the given information.

Key Concepts

Conservation of Linear Momentum

In any collision where external forces are negligible, the total linear momentum of the system is conserved. This means that the vector sum of momenta before and after the collision remains the same. When dealing with objects moving in directions that are not aligned, momentum must be conserved separately in each coordinate direction (such as the x? and y-axes), which helps in determining unknown direction components of velocity after the collision.

Vector Decomposition of Momentum

Momentum is a vector quantity, so when objects deflect into different directions after a collision, their momentum must be broken into components along the x and y axes (or any chosen coordinate system). This decomposition allows the use of conservation equations in each independent direction, facilitating the determination of unknown directions of motion even when some speeds are not known.

Indeterminacy Due to Missing Energy Information

While momentum conservation provides directional and component information, the magnitude (speed) of the objects after the collision may depend on additional factors such as the elasticity of the collision. Without knowing whether the collision is perfectly elastic, partially elastic, or inelastic (i.e., without information on energy conservation or the coefficient of restitution), the speed of one object cannot be uniquely determined since kinetic energy may not be conserved even though momentum is.

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