Detail as below Use the following Active Figure representing a collision between

Question

Detail as below

Use the following Active Figure representing a collision between two particles to complete this exercise. An object of mass m1=10 velocity Vi = -11 m/s crashes into another object of mass2 = 13 kg and velocity V2 = 6 m/s. The two particles stick together as a result of the collision. Determine the velocity Vf of the objects after collision. Determine the change in total mechanical energy. RECOGNIZE THE PRINCIPLES The two particles with the masses given approach each other and then stick together, continuing their motion as if they were now one particle whose mass is the sum of the previous two masses. The problem is concerned with a collision that occurs in one dimension. The system consists of the two objects and the only forces affecting the motion are the internal forces between them during collision. IDENTIFY THE RELATIONSHIPS Determine the velocity after collision. Because no external force acts, the collision does not change the total momentum of the system of two particles. We set the total momentum before collision to the total momentum afterward. mivu + m2v2i =(m1 + m2)vf The final velocity of the two objects joined together is given by the previous momentum conservation equation Vf =m1v1 + m2v2i/m1 + m2 Using the values of this exercise, we get v f=(10 kg)(-11 m/s) + (13 kg)(6 m/s) / (10 kg) + (13 kg) = m/s IDENTIFY THE RELATIONSHIPS (B) Determine the change in total mechanical energy. While the forces that the two objects exert on each other cannot change their total momentum, they can change the total kinetic energy in an inelastic collision such as the one being considered. All of the mechanical energy in the problem is kinetic energy. The change in kinetic energy is then Delta KE =1/2 (mi+ m2)vf2 - (1/2m1v1i2 + 1/2m2v2i2). At this point, we can either substitute Equation (1) for Vf into the expression for AKE and simplify the expression before doing the calculation, or we could substitute in the numerical values to evaluate Delta KE. The first procedure is more useful in exploring how the loss in kinetic energy depends on the velocities and masses of the colliding objects, while the second is easier if only a numerical answer is needed. Following the first procedure leads to Delta KE = -1/2(m1m2/m1+m2)(v1i - v2i)2. This could be used to calculate the same numerical answer, but it also shows various features that you might examine in the simulation. For example, notice that the change in kinetic energy depends on velocity only through the relative initial velocity, or difference in initial velocity (vy - /), of the two particles. Directly substituting in numerical values gives Delta KE = 1/2(10 kg + 13 kg)(vf m/s)2 -[1/2 [y(10 kg)(-ll m/s)2 + 1/2(13 kg)(6 m/s)2] = J. For a given initial velocity with the second object initially at rest, does the solution to this problem suggest that a large ratio of mi to m2 or a small ratio produces the largest possible final velocity? When the two objects initially have equal and oppositely-directed velocities, what does the momentum conservation equation indicate will happen? If the two particles are of equal mass, and the second particle is initially stationary, what simple relation applies between the initial velocity and the final velocity, and why?

Explanation / Answer

Inelastic collision, after collision both the mass move together.

Vf = -1.39 m/s

change in KE = -816.78 N

for the given inital velocity with the second object initially at rest, the solution of this problem will change,

large m1 to m2 ratio will produce largest possible final velocity.

when two objects initally have equal and oppositely directed velocities, the conservation of momentum will be m1*v + m2*(-v) = (m1 + m2) Vf

so Vf = (m1 - m2) * v/(m1+m2)

if two particles are of equal mass and the second particle is intially stationary,

the conservation of momentum equation will be simplified as

m * v + m * 0 = (m+m)*vf

m*v = 2m * Vf

Vf = v/2

the equation is simpler because the mass term (equal mass) is getting cancelled. and zero second object velocity.

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