Elastic and Inelastic Collisions

FIELDS OF STUDY: Classical Mechanics

ABSTRACT: Whether a car hits a truck, a train hits a stalled car, or one billiard ball hits another, collisions are inevitable in the world. Sometimes collisions conserve energy, and sometimes they dissipate energy. If energy is conserved, the collision is said to be an elastic collision; if energy is lost, the collision is either inelastic or completely inelastic. Regardless of the type of collision, one thing is certain: momentum is always conserved.

Principal Terms

• collision: an interaction in which two or more bodies come into contact and briefly exert force on each other.
• conservation of momentum: in physics, the principle that the total momentum in a closed system is always constant.
• dissipation: the irreversible loss of energy from a system.
• kinetic energy: the energy a body contains as a result of its motion.

Crashing Objects

Every time somebody walks through a room, millions of collisions are happening every second. Not only is the person’s body colliding with the surrounding air molecules, but his or her feet are colliding with the floor with every step. In each of the countless collisions that take place every day, numerous and varied forces act on the colliding objects as they make contact with each other. Some of these forces are nonconservative forces, resulting in the dissipation of energy, especially kinetic energy.

One such nonconservative force is friction. Although friction causes some loss of energy in every collision, it would be impossible to move without it. Every time a person takes a step, he or she is pushing back against the floor, creating friction. Isaac Newton’s (1642–1727) third law of motion states that for every action, there is a reaction of equal magnitude and opposite direction. So as the person’s foot exerts a friction force backward on the floor, the floor exerts an answering friction force on the foot that propels the person forward. Without friction, people would not be able to walk.

Nonconservative forces are almost inescapable on Earth. In most collisions between large bodies, energy is lost through these forces. When a person claps his or her hands and then keeps them together, all of the kinetic energy from the movement is lost. This energy does not disappear from the universe; it is transformed into other kinds of energy. Some of the kinetic energy becomes thermal energy, which is why clapping repeatedly for more than a few seconds causes a person’s hands to feel warm. Some of it becomes sound energy, which is why clapping makes a sharp sound.

Due to the abundance of nonconservative forces, most collisions between large bodies are inelastic collisions. Inelasticity is the property that allows objects to be deformed by a collision, while elasticity is the property that allows objects to return to their original shape. In an elastic collision, the kinetic energy of the system—in this case, the colliding objects—is conserved. A good example of an elastic collision is one in which two objects of equal mass collide and interchange velocities. In an inelastic collision, the kinetic energy is not conserved, and part of it is lost to the surrounding environment. This is the most common type of collision. A good example of an inelastic collision is a car crash.

A collision in which the maximum possible kinetic energy is lost is said to be perfectly or completely inelastic. In a completely inelastic collision, the colliding objects stick together. An arrow hitting a target and a tennis ball sticking to a Velcro surface are good examples of completely inelastic collisions.

Regardless of whether a collision is elastic or inelastic, momentum is always conserved. Linear momentum (p) is defined as the product of an object’s mass (m) and its velocity (v). Momentum and velocity are represented by bolded variables to indicate that they are vector quantities; that is, they have both a magnitude and a direction. (The counterpart to a vector quantity is a scalar quantity, such as mass, which has magnitude but no innate direction.) Due to this, one must consider the momentum of an object by the direction of its motion.

Conservation of momentum states that the initial overall momentum of a system (p_i) is equal to the final overall momentum of the system (p_f). These values represent the sum of all initial and final momenta, respectively, of each individual object in the system. To find the initial momentum, add up the individual initial momenta of each of the objects. The same procedure is used to find the final momentum.

Inelastic and Completely Inelastic Collisions

In an inelastic or completely inelastic collision, some kinetic energy is lost. Both types of collision are treated the same way, and momentum is conserved in both cases. The only difference between the two is that in a completely inelastic collision, the colliding objects stick together.

Imagine a car with a mass of 2,000 kilograms (kg) traveling east at 25 meters per second (m/s). The car hits a truck stopped at a red light. The truck has a mass of 4,000 kg. After the collision, the car and truck are stuck together and moving at the same velocity. The car and the truck together constitute a system, so their shared final velocity can be calculated by first determining the total momentum of the system (dealing with magnitude only and ignoring direction):

p = mv

Initially the truck is not moving, meaning that its velocity is 0 m/s and it has no momentum. After the collision, because the car and the truck stick together, their velocities are the same. Thus, the total momentum of the system before the collision is calculated as follows, using measurements for the car:

p_i = m_cv_c,i

p_i = (2,000 kg)(25 m/s)

p_i = 50,000 kg·m/s

Because momentum is conserved, p_f must also equal 50,000 kg·m/s. This information can be used to calculate the car and truck’s final velocity. Both are moving at the same velocity, so their velocities can be represented by the same variable (v_f).

p_f = m_cv_f + m_tv_f

p_f = (m_c + m_t)v_f

50,000 kg·m/s = (2,000 kg + 4,000 kg)(v_f)

8.33 m/s = v_f

The final velocity of both vehicles is approximately 8.33 m/s (rounded).

Because this is a completely inelastic collision, energy is lost, although momentum is not. To calculate the change in energy, find both the initial and the final kinetic energy of the system. Kinetic energy is a function of an object’s mass (m) and the square of its velocity (v):

The initial and final kinetic energies of the system are calculated as follows:

The initial kinetic energy is 625,000 joules (J). (A joule is equal to 1 kg·m²/s².) The final kinetic energy is 208,166.7 J. The energy lost in the collision is equal to the change in kinetic energy:

ΔK = K_f − K_i

ΔK = 208,166.7 J − 625,000 J

ΔK = −416,833.3 J

A total of 416,833.3 J of energy is lost during this completely inelastic collision.

While the car and the truck are colliding, they are exerting forces on each other: the car hits the truck, and the truck hits back. These forces are equal in magnitude but opposite in direction. As a result, both the car and the truck experience the same change in momentum. Because momentum is a function of mass and velocity, and the truck has a greater mass than the car, this means that the change in the truck’s velocity will be smaller than the change in the velocity of the car.

Now imagine that instead of the two vehicles sticking together after the collision, the truck moves forward at a velocity of 11 m/s and the car moves forward at 3 m/s. Although the collision is not completely inelastic, some energy is still lost. The initial kinetic energy is the same as in the previous example (625,000 J). The final kinetic energy is calculated as follows:

Subtract the initial kinetic energy from the final kinetic energy:

Only 374,000 J of energy are lost in this collision. Less energy is lost than in the previous scenario because the collision is not completely inelastic. As in the completely inelastic collision, the truck exerts a force on the car that is equal in magnitude and opposite in direction to the force of the car hitting the truck. And again, due to momentum being dependent on mass, the truck suffers the smaller change in velocity.

Elastic Collisions

In an elastic collision, energy is conserved. This type of collision is extremely rare on Earth, because nonconservative forces such as friction almost always play a role. However, in some cases, the effects of these forces are small enough that they can be ignored.

One such case is that of balls bouncing off one another. Imagine a ball with a mass of 10 kg, moving at 5 m/s along an imaginary x axis in the positive (+x) direction, from left to right. This ball hits another, stationary ball of equal mass. The collision happens off-center, so that after the collision, both balls are moving diagonally: while both are now traveling in +x direction at a velocity of 2.5 m/s, one ball is simultaneously moving up the y axis, in the positive (+y) direction, at a velocity of 2.5 m/s, and the other ball is moving down the y axis, in the negative (−y) direction, also at 2.5 m/s. Because momentum is a vector quantity and this case deals with multiple directions, each axis has to be considered separately.

Only one ball is moving before the collision, so the initial momentum of the system consists only of the momentum of that one ball, and only in +x direction. The ball’s initial momentum is calculated as follows:

p_i,x = m₁v_1,i,x

p_i,x = (10 kg)(5 m/s)

p_i,x = 50 kg·m/s

After the collision, both balls are moving in +x direction with a velocity of 2.5 m/s. The final momentum in this direction is:

p_f,x = m₁v_1,f,x + m₂v_2,f,x

p_f,x = (10 kg)(2.5 m/s) + (10 kg)(2.5 m/s)

p_f,x = 25 kg·m/s + 25 kg·m/s = 50 kg·m/s

Both the initial and final momenta in +x direction are 50 kg·m/s, so momentum along this axis is conserved.

Now consider the y axis. The initial momentum along this axis is 0 kg·m/s, because before the collision, neither ball is moving in either +y or −y direction. After the collision, one ball moves in +y direction at a velocity of 2.5 m/s, and the other ball moves in −y direction at the same velocity. Because momentum is a vector quantity, the momentum of the ball moving in +y direction is positive, while the momentum of the ball moving in −y is negative. Thus, the total combined final momentum in along the y axis is:

p_f,y = m₁v_1,f,x + (−m₂v_2,f,x)

p_f,y = (10 kg)(2.5 m/s) − (10 kg)(2.5 m/s)

p_f,y = 25 kg·m/s − 25 kg·m/s = 0 kg·m/s

The final combined momentum is also 0 kg·m/s. Momentum along the y axis is also conserved.

Sample Problem

In a game of pool, one player hits the cue ball, which has a mass of 0.26 kg, toward the red three ball, which has a mass of 0.17 kg. The cue ball hits the stationary red ball at a velocity of 1.5 m/s in an elastic collision. The red ball moves to the left at approximately 1.57 m/s, while the cue ball moves to the right at 0.8 m/s. How much energy is transferred to the red ball during the collision?

Answer:

This is an elastic collision, so there is no loss of kinetic energy from the system. Thus, the system’s initial kinetic energy and its final kinetic energy are the same:

K_i = K_f

The total kinetic energy of the system consists of the combined individual kinetic energies of the cue ball and the red ball:

K_c,i + K_r,i = K_c,f + K_r,f

Because the red ball is stationary at first, it has no initial kinetic energy, so K_r,i = 0. Solve for K_r,f, then substitute the definition of kinetic energy for K_c,i and K_c,f and simplify:

Plug in the known values for each variable and solve:

The final kinetic energy of the red ball is 0.2093 J. Because it had no initial kinetic energy, that is also the amount of energy that is transferred to it during the collision.

Collisions All Around

Because nonconservative forces mean that most collisions on Earth result in a loss of energy, solving problems based on conservation of energy is not a very useful technique. However, using the principle of conservation of momentum will always lead to the correct answer. Collisions happen constantly, whether they consist of a person walking and colliding with the surrounding air molecules or two cars colliding on the highway. Even in recreational activities, proper collision calculations can be made by using conservation of momentum.

In particle physics, collisions are extremely important. Particle accelerators, such as the Large Hadron Collider, work by crashing particles into other particles. The resulting elastic collisions break the particles apart into their respective components, allowing scientists to study the makeup of matter itself.

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