Math / Algebra

QuestionFor an elastic collision, both energy and momentum are conserved for a system. In this particular elastic collision, assume that it is a head-on collision in one dimension. The first object has a mass of $m_{1}$ and initial speed of $v_{1 i}$ , and the second object has a mass of $m_{2}$ and is initially at rest. Follow these steps to find the final speeds $v_{1 f}$ and $v_{2 f}$ of the two objects after the elastic collision. Record your answers in your lab notebook before turning in this worksheet.
1. Write an expression for the initial kinetic energy of (2 pts): a. Object 1 b. Object 2
2. Write an expression for the final kinetic energy of (2 pts): a. Object 1 b. Object 2
3. Write an equation for conservation of kinetic energy for this collision. (2 pts)
4. Write an expression for the initial momentum of (2 pts): a. Object 1 b. Object 2

Studdy Solution

STEP 1

1. The collision is elastic, meaning both kinetic energy and momentum are conserved.
2. The collision is head-on and occurs in one dimension.
3. The second object is initially at rest.

STEP 2

1. Write expressions for the initial kinetic energy of both objects.
2. Write expressions for the final kinetic energy of both objects.
3. Write the equation for conservation of kinetic energy.
4. Write expressions for the initial momentum of both objects.

STEP 3

The initial kinetic energy of object 1 is given by the formula for kinetic energy: $KE_{1i} = \frac{1}{2} m_1 v_{1i}^2$

STEP 4

Since object 2 is initially at rest, its initial kinetic energy is: $KE_{2i} = \frac{1}{2} m_2 \times 0^2 = 0$

STEP 5

The final kinetic energy of object 1 is: $KE_{1f} = \frac{1}{2} m_1 v_{1f}^2$

STEP 6

The final kinetic energy of object 2 is: $KE_{2f} = \frac{1}{2} m_2 v_{2f}^2$

STEP 7

The conservation of kinetic energy for this collision is expressed as: $KE_{1i} + KE_{2i} = KE_{1f} + KE_{2f}$
Substituting the expressions from previous steps: $\frac{1}{2} m_1 v_{1i}^2 + 0 = \frac{1}{2} m_1 v_{1f}^2 + \frac{1}{2} m_2 v_{2f}^2$

STEP 8

The initial momentum of object 1 is given by: $p_{1i} = m_1 v_{1i}$

STEP 9

Since object 2 is initially at rest, its initial momentum is: $p_{2i} = m_2 \times 0 = 0$
The expressions for initial kinetic energy, final kinetic energy, conservation of kinetic energy, and initial momentum have been successfully derived.

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Studdy Solution

STEP 1

1. The collision is elastic, meaning both kinetic energy and momentum are conserved.
2. The collision is head-on and occurs in one dimension.
3. The second object is initially at rest.

STEP 2

1. Write expressions for the initial kinetic energy of both objects.
2. Write expressions for the final kinetic energy of both objects.
3. Write the equation for conservation of kinetic energy.
4. Write expressions for the initial momentum of both objects.

STEP 3

The initial kinetic energy of object 1 is given by the formula for kinetic energy: $KE_{1i} = \frac{1}{2} m_1 v_{1i}^2$

STEP 4

Since object 2 is initially at rest, its initial kinetic energy is: $KE_{2i} = \frac{1}{2} m_2 \times 0^2 = 0$

STEP 5

The final kinetic energy of object 1 is: $KE_{1f} = \frac{1}{2} m_1 v_{1f}^2$

STEP 6

The final kinetic energy of object 2 is: $KE_{2f} = \frac{1}{2} m_2 v_{2f}^2$

STEP 7

The conservation of kinetic energy for this collision is expressed as: $KE_{1i} + KE_{2i} = KE_{1f} + KE_{2f}$
Substituting the expressions from previous steps: $\frac{1}{2} m_1 v_{1i}^2 + 0 = \frac{1}{2} m_1 v_{1f}^2 + \frac{1}{2} m_2 v_{2f}^2$

STEP 8

The initial momentum of object 1 is given by: $p_{1i} = m_1 v_{1i}$

STEP 9

Since object 2 is initially at rest, its initial momentum is: $p_{2i} = m_2 \times 0 = 0$
The expressions for initial kinetic energy, final kinetic energy, conservation of kinetic energy, and initial momentum have been successfully derived.

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Studdy Solution

STEP 1

1. The collision is elastic, meaning both kinetic energy and momentum are conserved.2. The collision is head-on and occurs in one dimension.3. The second object is initially at rest.

STEP 2

1. Write expressions for the initial kinetic energy of both objects.2. Write expressions for the final kinetic energy of both objects.3. Write the equation for conservation of kinetic energy.4. Write expressions for the initial momentum of both objects.

STEP 3

The initial kinetic energy of object 1 is given by the formula for kinetic energy: KE1i=12m1v1i2 KE_{1i} = \frac{1}{2} m_1 v_{1i}^2 KE1i​=21​m1​v1i2​

STEP 4

Since object 2 is initially at rest, its initial kinetic energy is: KE2i=12m2×02=0 KE_{2i} = \frac{1}{2} m_2 \times 0^2 = 0 KE2i​=21​m2​×02=0

STEP 5

The final kinetic energy of object 1 is: KE1f=12m1v1f2 KE_{1f} = \frac{1}{2} m_1 v_{1f}^2 KE1f​=21​m1​v1f2​

STEP 6

The final kinetic energy of object 2 is: KE2f=12m2v2f2 KE_{2f} = \frac{1}{2} m_2 v_{2f}^2 KE2f​=21​m2​v2f2​

STEP 7

STEP 8

The initial momentum of object 1 is given by: p1i=m1v1i p_{1i} = m_1 v_{1i} p1i​=m1​v1i​

STEP 9

Since object 2 is initially at rest, its initial momentum is: p2i=m2×0=0 p_{2i} = m_2 \times 0 = 0 p2i​=m2​×0=0 The expressions for initial kinetic energy, final kinetic energy, conservation of kinetic energy, and initial momentum have been successfully derived.

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1. The collision is elastic, meaning both kinetic energy and momentum are conserved.
2. The collision is head-on and occurs in one dimension.
3. The second object is initially at rest.

1. Write expressions for the initial kinetic energy of both objects.
2. Write expressions for the final kinetic energy of both objects.
3. Write the equation for conservation of kinetic energy.
4. Write expressions for the initial momentum of both objects.

The initial kinetic energy of object 1 is given by the formula for kinetic energy: $KE_{1i} = \frac{1}{2} m_1 v_{1i}^2$

Since object 2 is initially at rest, its initial kinetic energy is: $KE_{2i} = \frac{1}{2} m_2 \times 0^2 = 0$

The final kinetic energy of object 1 is: $KE_{1f} = \frac{1}{2} m_1 v_{1f}^2$

The final kinetic energy of object 2 is: $KE_{2f} = \frac{1}{2} m_2 v_{2f}^2$

The initial momentum of object 1 is given by: $p_{1i} = m_1 v_{1i}$

Since object 2 is initially at rest, its initial momentum is: $p_{2i} = m_2 \times 0 = 0$
The expressions for initial kinetic energy, final kinetic energy, conservation of kinetic energy, and initial momentum have been successfully derived.