Have you ever played pool or seen someone playing pool? As you play, you hit a ball at a billiard table and that ball hits the other balls on the table, causing them to move as well. A similar situation can be seen while playing volleyball. When you get ready to play, you practice with the ball. When you hit it to the ground, the ball bounces back. These situations are examples of elastic collision.
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Jetzt kostenlos anmeldenHave you ever played pool or seen someone playing pool? As you play, you hit a ball at a billiard table and that ball hits the other balls on the table, causing them to move as well. A similar situation can be seen while playing volleyball. When you get ready to play, you practice with the ball. When you hit it to the ground, the ball bounces back. These situations are examples of elastic collision.
Elastic collisions are collisions in which the total kinetic energy stays the same before and after the collision. However, the kinetic energies of individual objects can change.
Elastic Collision: A collision in which the total kinetic energy of the objects stays the same throughout the collision.
As in the example of playing pool, a moving ball can hit a stationary ball and make it move as well. A moving ball has a certain velocity and thus also has kinetic energy. When it hits a stationary ball, it transfers some of its energy to the other ball. This means that the kinetic energy of the moving ball decreases, and the kinetic energy of the stationary ball increases.
When both objects are moving and hit each other, an elastic collision can still happen. The total kinetic energy and momentum will still be conserved, but the formulas for the final velocities will differ from the stationary target case.
When an object with mass \(m_1\) and an initial velocity \(V_{1i}\) hits a stationary ball with mass \(m_2\), the object with mass \(m_1\) has a final velocity \(V_{1f}\) and the object with mass \(m_2\) has a final velocity \(V_{2f}\). The net linear momentum is conserved, that's why the total momentum stays the same before and after the collision.
The equation for the linear momentum conservation can be written as:
Equation 1: \(m_1V_{1i}=m_1V_{1f}+m_2V_{2f}\)
The equation for the kinetic energy conservation can be written as:
Equation 2: \(\frac 1 2 m_1(V_{1i})^2=\frac 1 2 m_1(V_{1f})^2+\frac 1 2m_2(V_{2f})^2\)
To find the final velocities, we can do some algebra. We can convert the momentum equation to:
Equation 3: \(m_1V_{1i}-m_1V_{1f}=m_2V_{2f}\)
Since \(m_1\) is common on the left side, we can rearrange the equation:
Equation 4: \(m_1(V_{1i}-V_{1f})=m_2V_{2f}\)
We can rearrange the energy equation as well. First of all, we can multiply the equation by 2.
Equation 5: \(m_1(V_{1i})^2=m_1(V_{1f})^2+m_2(V_{2f})^2\)
We can move \(m_1(V_{1f})^2\) to the left side.
Equation 6: \(m_1(V_{1i})^2-m_1(V_{1f})^2=m_2(V_{2f})^2\)
Since \(m_1\) is common on the left side, we can now rearrange the equation:
Equation 7: \(m_1((V_{1i})^2-(V_{1f})^2)=m_2(V_{2f})^2\)
Because we are subtracting two items that are in the square form, we can rewrite it as:
Equation 8: \(m_1(V_{1i}+V_{1f})(V_{1i}-V_{1f})=m_2(V_{2f})^2\)
After dividing Equation 8 by Equation 4 and then rearranging, we can find the elastic collision formulas for the stationary target case.
$$V_{1f}=\frac {m_1-m_2}{m_1+m_2}V_{1i}$$
$$V_{2f}=\frac {2m_1}{m_1+m_2}V_{1i}$$
When an object with mass \(m_1\) and an initial velocity \(V_{1i}\) hits a ball with mass \(m_2\) and an initial velocity \(V_{2i}\), the object with mass \(m_1\) has a final velocity \(V_{1f}\) and the object with mass \(m_2\) has a final velocity \(V_{2f}\).
We can write the momentum conservation equation as:
Equation 10: \(m_1V_{1i}+m_2V_{2i}=m_1V_{1f}+m_2V_{2f}\)
We can write the kinetic energy conservation as:
Equation 11: \(\frac 12m_1(V_{1i})^2+\frac 12m_2(V_{2i})^2=\frac 12m_1(V_{1f})^2+\frac 12m_2(V_{2f})^2\)
We can rearrange Equation 10:
Equation 12: \(m_1V_{1i}-m_1V_{1f}=m_2V_{2f}-m_2V_{2i}\)
Equation 13: \(m_1(V_{1i}-V_{1f})=-m_2(V_{2i}-V_{2f})\)
Also, we can rearrange Equation 11:
Equation 14: \(m_1(V_{1i}-V_{1f})(V_{1i}+V_{1f})=-m_2(V_{2i}-V_{2f})(V_{2i}+V_{2f})\)
We can divide Equation 14 by Equation 13 and rearrange it to get the formulas for the final velocities of the objects:
$$V_{1f}=\frac{m_1-m_2}{m_1+m_2}V_{1i}+\frac{2m_2}{m_1+m_2}V_{2i}$$
$$V_{2f}=\frac{2m_1}{m_1+m_2}V_{1i}+\frac{m_2-m_1}{m_1+m_2}V_{2i}$$
For the stationary target case, our velocity equations are:
$$V_{1f}=\frac {m_1-m_2}{m_1+m_2}V_{1i}$$
$$V_{2f}=\frac {2m_1}{m_1+m_2}V_{1i}$$
For the moving target case, our final velocity equations are different:
$$V_{1f}=\frac{m_1-m_2}{m_1+m_2}V_{1i}+\frac{2m_2}{m_1+m_2}V_{2i}$$
$$V_{2f}=\frac{2m_1}{m_1+m_2}V_{1i}+\frac{m_2-m_1}{m_1+m_2}V_{2i}$$
A ball with a mass of 2 kg is moving to the right with a velocity of 4 m/s and hits a stationary ball with a mass of 1 kg at the billiard table. What are the final velocities of the balls?
Solution:
To find the final velocities, we can use Equation 9. As given in the example, \(V_{1i}=4 \frac ms\) and \(m_1=2 kg\). We can insert these values into the equation.
$$V_{1f}=\frac {2 kg - 1 kg}{2 kg + 1 kg}\times 4 \frac ms = \frac 1 3 \times 4 \frac ms = \frac 4 3 \frac ms$$
$$V_{2f}=\frac {2 \times 2 kg}{2 kg + 1 kg}\times 4 \frac ms = \frac 4 3 \times 4 \frac ms = \frac {16} 3 \frac ms$$
A ball with a mass of 6 kg and an initial velocity of 4 m/s hits another ball with a mass of 4 kg and an initial velocity of 2 m/s. They both are moving to the right. What are their final velocities after the collision?
Solution:
We can use our formula from above to find the final velocities. Here, \(m_1=6 kg\) and \(m_2=4 kg\), \(V_{1i} = 4 \frac ms\) and \(V_{2i}=2 \frac ms\).
$$V_{1f}=\frac{6 kg-4 kg}{6 kg+4 kg}\times 4 \frac ms + \frac{2\times 4 kg}{6 kg+4 kg}\times 2 \frac ms$$
$$V_{1f}= \frac 2{10}\times 4+ \frac 8{10}\times 2 = \frac 8{10}+\frac{16}{10}=\frac{24}{10}\Rightarrow 2.4 \frac ms$$
$$V_{2f}=\frac{2\times6 kg}{6 kg + 4 kg}\times 4\frac ms + \frac{4 kg - 6 kg}{6 kg + 4 kg}\times 2\frac ms$$
$$V_{2f} = \frac {12}{10}\times 4 + \frac{-2}{10}\times 2 = \frac{48}{10}-\frac 4{10} = \frac{44}{10}\Rightarrow 4.4 \frac ms$$
Playing pool and other situations involving bouncing balls can be examples of elastic collision.
Elastic collisions are collisions in which the total kinetic energy stays the same before and after the collision.
Yes.
While energy is conserved in elastic collisions, it is not conserved in inelastic collisions.
Yes.
What is an elastic collision?
Elastic collisions are collisions in which the mechanical energy is conserved.
Is momentum conserved after an elastic collision?
The total momentum is conserved after the collision.
A ball with a mass of \(4 \,\mathrm{kg}\) is moving with a velocity of \(8 \,\mathrm{\frac{m}{s}}\) to the right and hits a stationary ball with a mass of \(2 \,\mathrm{kg}\) at the billiard table. What are the final velocities of the balls?
\(v_{1f}=\frac 83 \mathrm{\frac ms}\)
\(v_{1f}=\frac {32}3 \mathrm{\frac ms}\)
A ball with a mass of \(1 kg\) is moving with \(3 m/s\) to the right and hits a stationary ball with a mass of \(4 kg\) at the billiard table. What are the final velocities of the balls?
\(V_{1f}=\frac{-9}5 \frac ms\)
\(V_{2f}=\frac 65 \frac ms\)
A ball with a mass of \(5 kg\) is moving with \(2 m/s\) to the right and hits a stationary ball with a mass of \(1 kg\).
What are the final velocities of the balls?
\(V_{1f}=\frac 43 \frac ms\)
\(V_{2f}=\frac{10}3 \frac ms\)
A ball with a mass of \(4 kg\) is moving with an initial velocity to the right and hits a stationary ball with a mass of \(1 kg\). The final velocity of the stationary ball is \(2 m/s\). Then what is the initial velocity of the moving ball?
\(V_{1i}=\frac 54 \frac ms\)
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