Elastic Collision Problem, WOULD REALLY APPRECIATE HELP

In summary, the textbook solved the problem using the conservation of momentum equation and the characteristic of elastic collisions where the relative velocities before and after the collision are equal and opposite. This is a simpler way to solve the problem compared to using the conservation of kinetic energy equation. The textbook's approach can be proven by solving the general case of an elastic collision using the conservation of energy equation and finding that the relationship between the relative velocities holds true.
  • #1
starburst
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0
Hi everyone. I would really appreciate if I could get some help with this problem elastic collision problem. I will provide the question and the solution as shown in my textbook. What I would like to know if specifically how they used the equation, V2ix-V1ix = -(V2fx-V1fx), and why they arranged it the way they did. Is there perhaps a different way to solve the problem? Thank you very much!

Homework Statement



At a Route 3 highway on-ramp, a car of mass 1.50e3 kg is stopped at a stop sign, waiting for a break in traffic before merging with the cars on the highway. A pickup of mass 2.00e3 kg comes up from behind and hits the stopped car. Assuming the collision is elastic, how fast was the pickup

Homework Equations



M1V1x + M2V2iX=M1Vfx + M2V2fx

V2ix-V1ix = -(V2fx-V1fx)

3. The solution in the book

"From conservation of momentum: M2V2i=M1Vf + M2V2f (1) because the intital velocity of mass 1 is 0 m/s

The collision is elastic, so the relative velocity after the collision is equal and opposite to the relative velocity before the collision: V2i = -(V2f-V1f) (2)

We watn to solve these two equations for V2i, so we can eliminate V2f. Multiplying eq. (2) through by M2 and rearanging yields: M2V2i = M2V1f - M2V2f (3)

Adding eqs. (1) and (3) gives: 2*M2V2i = (M1+M2)V1f (4)

Finally we solve eq. (4) for V2i: V2i = M1+M1/2M2 * V1f = 1500 kg + 2000 kg/4000 kg * 20.0 m/s = 17.5 m/s "



Could someone please explain to me how the textbook solved it this way? Or how it can be solved in a simpler way? Thank you very much!
 
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  • #2
starburst said:
Could someone please explain to me how the textbook solved it this way? Or how it can be solved in a simpler way? Thank you very much!

It's really a very simple way to get the required two equations in two unknowns that are required to solve for the two final velocities. It relies on a (provable) characteristic of elastic collisions whereby the relative velocities of the two objects, before and after collision, are of equal magnitude but opposite sign.

Another way to solve the problem is to use conservation of kinetic energy as the second equation (KE is conserved in elastic collisions). But this introduces the squares of the velocities into the mix, which turn into square roots of expressions, and are a bit harder to work with.

If you want you can prove the relative velocity relationship by solving the general case elastic collision using the conservation of energy approach to yield expressions for the final velocities, and then use them to find the final relative velocity. You will then find that the relationship is true in general.
 

Related to Elastic Collision Problem, WOULD REALLY APPRECIATE HELP

What is an elastic collision?

An elastic collision is a type of collision in which the total kinetic energy of the system is conserved. This means that the energy before the collision is equal to the energy after the collision.

What is the difference between an elastic and an inelastic collision?

In an elastic collision, the total kinetic energy is conserved, while in an inelastic collision, some of the kinetic energy is lost to other forms of energy, such as heat or sound.

What is the equation for calculating the final velocities in an elastic collision?

The equation for calculating the final velocities in an elastic collision is: v1f = (m1 - m2) / (m1 + m2) * v1i + (2m2) / (m1 + m2) * v2iand v2f = (2m1) / (m1 + m2) * v1i + (m2 - m1) / (m1 + m2) * v2i

What are some real-world examples of elastic collisions?

Some examples of elastic collisions in everyday life include billiard balls colliding on a pool table, bumper cars colliding in an amusement park, or a tennis ball bouncing off a racket.

What happens in an elastic collision if one of the objects is stationary?

If one of the objects is stationary in an elastic collision, the final velocities of the two objects will be equal in magnitude, but opposite in direction. This is because the stationary object has no initial velocity, so all of the initial kinetic energy is transferred to the moving object.

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