The momentum of a system before a collision is [tex]$2.4 \times 10^3 kilogram$[/tex] meters/second in the [tex]$x$[/tex]-direction and [tex]$3.5 \times 10^3 kilogram$[/tex] meters/second in the [tex]$y$[/tex] direction. What is the magnitude of the resultant momentum after the collision if the collision is inelastic?
A. [tex]$1.7 \times 10^3$[/tex] kilogram meters/second
B. [tex]$2.1 \times 10^3$[/tex] kilogram meters/second
C. [tex]$3.4 \times 10^3$[/tex] kilogram meters/second
D. [tex]$4.2 \times 10^3 kilogram$[/tex] meters/second
E. [tex]$5.7 \times 10^3$[/tex] kilogram meters/second
Answer (2)
Calculate the magnitude of the total momentum before the collision using the Pythagorean theorem: p = p x 2 + p y 2 .
Substitute the given values: p = ( 2.4 × 1 0 3 ) 2 + ( 3.5 × 1 0 3 ) 2 .
Calculate the magnitude: p = 4.2438 × 1 0 3 k g ⋅ m / s .
Round to one decimal place: The magnitude of the resultant momentum after the collision is 4.2 × 1 0 3 k g ⋅ m / s .
Explanation
Understanding the Problem We are given the momentum of a system before a collision in the x-direction ( p x = 2.4 × 1 0 3 k g "." m / s ) and in the y-direction ( p y = 3.5 × 1 0 3 k g "." m / s ). We need to find the magnitude of the resultant momentum after the collision, knowing that the collision is inelastic.
Momentum Conservation Since momentum is conserved in all collisions (elastic or inelastic), the total momentum before the collision is equal to the total momentum after the collision. Therefore, we need to calculate the magnitude of the total momentum before the collision.
Calculating Total Momentum To find the magnitude of the total momentum before the collision, we use the Pythagorean theorem: p = p x 2 + p y 2 where p x and p y are the momenta in the x and y directions, respectively.
Substituting Values and Calculating Substituting the given values, we have: p = ( 2.4 × 1 0 3 ) 2 + ( 3.5 × 1 0 3 ) 2 p = ( 5.76 × 1 0 6 ) + ( 12.25 × 1 0 6 ) p = 18.01 × 1 0 6 p = 4.2438 × 1 0 3 k g "." m / s Rounding to one decimal place, we get p = 4.2 × 1 0 3 k g "." m / s .
Final Answer Therefore, the magnitude of the resultant momentum after the collision is 4.2 × 1 0 3 k g "." m / s .
Examples
Consider a scenario where two cars collide at an intersection. If we know the momentum of each car before the collision in both the x and y directions, we can determine the total momentum of the system. Even if the collision is inelastic (meaning kinetic energy is not conserved), the total momentum of the system remains constant. This principle helps accident investigators reconstruct the collision and determine the velocities of the cars immediately after impact.
To find the magnitude of the momentum before an inelastic collision, we use the formula p = p x 2 + p y 2 . Calculating with the given values results in a magnitude of 4.2 × 1 0 3 kg m/s. Thus, the answer is option D: 4.2 × 1 0 3 kg m/s .
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