Enter any four inputs and click on Calculate button
How to use this Conservation of Momentum Calculator 🤔
There are input fields for Mass of object 1 \((m₁)\), Initial velocity of object 1 \((u₁)\), Final velocity of object 1 \((v₁)\), Mass of object 2 \((m₂)\), Initial velocity of object 2 \((u₂)\), and Final velocity of object 2 \((v₂)\). Enter any four inputs and click on Calculate button.
The calculator uses the \( m_1 \cdot u_1 + m_2 \cdot u_2 = m_1 \cdot v_1 + m_2 \cdot v_2 \) formula, substitues given values, and calcuates the missing value.
The missing value is calcuated and displayed in the input field. Also, the caculation is displayed under the input section.
What is Conservation of Momentum?
The principle of conservation of momentum states that in a closed system, where no external forces are acting, the total momentum before an interaction is equal to the total momentum after the interaction. This means that the momentum is conserved and does not change, even if the objects involved collide or interact.
Calculating Conservation of Momentum
The conservation of momentum can be calculated using the formula:
Consider that there is a head on collision of two objects. The collision is perfectly elastic. The two objects are having a mass of 10 kg and 8 kg with initial velocities of 20 m/s and 4 m/s respectively. What is the final velocity of first object when the final velocity of second object is 16 m/s?
Consider that there is a head on collision of two objects. The collision is perfectly elastic. The two objects are having a mass of 1 kg and 1 kg with initial velocities of 10 m/s and 5 m/s respectively. What is the final velocity of first object when the final velocity of second object is 10 m/s?
{
"topic": "conservation-of-momentum",
"input_types": [
"float",
"float",
"float",
"float",
"float",
"float"
],
"input_labels": [
"m₁",
"u₁",
"v₁",
"m₂",
"u₂",
"v₂"
],
"input_descriptions": [
"Mass of object 1",
"Initial velocity of object 1",
"Final velocity of object 1",
"Mass of object 2",
"Initial velocity of object 2",
"Final velocity of object 2"
],
"input_units": [
"kg",
"m/s",
"m/s",
"kg",
"m/s",
"m/s"
],
"input_values": [
"",
"",
"",
"",
"",
""
],
"formula_mathjax": "\\( m_1 \\cdot u_1 + m_2 \\cdot u_2 = m_1 \\cdot v_1 + m_2 \\cdot v_2 \\)",
"input_pre_msg": "Enter any four inputs and click on Calculate button",
"type": "Calculate",
"title": "Conservation of Momentum Calculator",
"description": "Calculate the final velocities or initial velocities of two objects in a closed system using our Conservation of Momentum Calculator. Ideal for physics students, teachers, and professionals.",
"category": "Kinematics",
"template": "physics",
"formulas": [
{
"parameters": [
"m₁",
"u₁",
"m₂",
"u₂",
"v₂"
],
"outputs": [
{
"label": "v₁",
"formula": "( m₁ * u₁ + m₂ * u₂ - m₂ * v₂ ) / m₁ ",
"formula_mathjax": "\\( \\frac{ m₁ \\times u₁ + m₂ \\times u₂ - m₂ \\times v₂ }{ m₁ } \\)"
}
],
"example_heading": "Consider that there is a head on collision of two objects. The collision is perfectly elastic. The two objects are having a mass of _m₁_ and _m₂_ with initial velocities of _u₁_ and _u₂_ respectively.<br>What is the final velocity of first object when the final velocity of second object is _v₂_?",
"examples": [
{
"m₁": 10,
"u₁": 20,
"m₂": 8,
"u₂": 4,
"v₂": 16,
"v₁": "10.4"
},
{
"m₁": 1,
"u₁": 10,
"m₂": 1,
"u₂": 5,
"v₂": 10,
"v₁": "5"
}
]
},
{
"parameters": [
"m₁",
"u₁",
"m₂",
"u₂",
"v₁"
],
"outputs": [
{
"label": "v₂",
"formula": "( m₁ * u₁ + m₂ * u₂ - m₁ * v₁ ) / m₂ ",
"formula_mathjax": "\\( \\frac{ m_1 \\cdot u_1 + m_2 \\cdot u_2 - m_1 \\cdot v_1 }{ m_2 } \\)"
}
]
},
{
"parameters": [
"m₁",
"v₁",
"m₂",
"v₂",
"u₂"
],
"outputs": [
{
"label": "u₁",
"formula": "( m₁ * v₁ + m₂ * v₂ - m₂ * u₂ ) / m₁ ",
"formula_mathjax": "\\( \\frac{ m_1 \\cdot v_1 + m_2 \\cdot v_2 - m_2 \\cdot u_2 }{ m_1 } \\)"
}
]
},
{
"parameters": [
"m₁",
"v₁",
"m₂",
"v₂",
"u₁"
],
"outputs": [
{
"label": "u₂",
"formula": "( m₁ * v₁ + m₂ * v₂ - m₁ * u₁ ) / m₂ ",
"formula_mathjax": "\\( \\frac{ m_1 \\cdot v_1 + m_2 \\cdot v_2 - m_1 \\cdot u_1 }{ m_2 } \\)"
}
]
}
],
"content": "<h2>What is Conservation of Momentum?</h2><p>The principle of conservation of momentum states that in a closed system, where no external forces are acting, the total momentum before an interaction is equal to the total momentum after the interaction. This means that the momentum is conserved and does not change, even if the objects involved collide or interact.</p><h2>Calculating Conservation of Momentum</h2><p>The conservation of momentum can be calculated using the formula:</p><p class=\"formula_highlite\">\\( m_1 \\cdot u_1 + m_2 \\cdot u_2 = m_1 \\cdot v_1 + m_2 \\cdot v_2 \\).</p><p>In this formula, the parameters are defined as follows:</p><ul><li><strong>m₁</strong>: The mass of the first object.</li><li><strong>u₁</strong>: The initial velocity of the first object before the interaction.</li><li><strong>m₂</strong>: The mass of the second object.</li><li><strong>u₂</strong>: The initial velocity of the second object before the interaction.</li><li><strong>v₁</strong>: The final velocity of the first object after the interaction.</li><li><strong>v₂</strong>: The final velocity of the second object after the interaction.</li></ul><br><h2>Input and Output Combinations of the Conservation of Momentum Calculator</h2><p>Here are the input combinations available for the calculator, along with the formulas it uses to calculate the corresponding outputs.</p><div class=\"formula_sets\"><div class=\"formula_set set_1\"><span class=\"heading_side\">Calculator Inputs</span><span class=\"set_n\">Set #1</span><ul class=\"given\"><li><span class=\"input\">m₁</span> Mass of object 1</li><li><span class=\"input\">u₁</span> Initial velocity of object 1</li><li><span class=\"input\">m₂</span> Mass of object 2</li><li><span class=\"input\">u₂</span> Initial velocity of object 2</li><li><span class=\"input\">v₂</span> Final velocity of object 2</li></ul><p>For these given inputs,</p><span class=\"heading_side\">Outputs, Formulas</span><p>We calculate the following outputs:</p><ul class=\"find\"><li><span class=\"output_label\">Final velocity of object 1</span><span class=\"output\">\\(v₁ = \\)\\( \\frac{ m₁ \\times u₁ + m₂ \\times u₂ - m₂ \\times v₂ }{ m₁ } \\)</span></li></ul><a class=\"button\" href=\"#example_1\">2 Examples</a></div><div class=\"formula_set set_2\"><span class=\"heading_side\">Calculator Inputs</span><span class=\"set_n\">Set #2</span><ul class=\"given\"><li><span class=\"input\">m₁</span> Mass of object 1</li><li><span class=\"input\">u₁</span> Initial velocity of object 1</li><li><span class=\"input\">m₂</span> Mass of object 2</li><li><span class=\"input\">u₂</span> Initial velocity of object 2</li><li><span class=\"input\">v₁</span> Final velocity of object 1</li></ul><p>For these given inputs,</p><span class=\"heading_side\">Outputs, Formulas</span><p>We calculate the following outputs:</p><ul class=\"find\"><li><span class=\"output_label\">Final velocity of object 2</span><span class=\"output\">\\(v₂ = \\)\\( \\frac{ m_1 \\cdot u_1 + m_2 \\cdot u_2 - m_1 \\cdot v_1 }{ m_2 } \\)</span></li></ul></div><div class=\"formula_set set_3\"><span class=\"heading_side\">Calculator Inputs</span><span class=\"set_n\">Set #3</span><ul class=\"given\"><li><span class=\"input\">m₁</span> Mass of object 1</li><li><span class=\"input\">v₁</span> Final velocity of object 1</li><li><span class=\"input\">m₂</span> Mass of object 2</li><li><span class=\"input\">v₂</span> Final velocity of object 2</li><li><span class=\"input\">u₂</span> Initial velocity of object 2</li></ul><p>For these given inputs,</p><span class=\"heading_side\">Outputs, Formulas</span><p>We calculate the following outputs:</p><ul class=\"find\"><li><span class=\"output_label\">Initial velocity of object 1</span><span class=\"output\">\\(u₁ = \\)\\( \\frac{ m_1 \\cdot v_1 + m_2 \\cdot v_2 - m_2 \\cdot u_2 }{ m_1 } \\)</span></li></ul></div><div class=\"formula_set set_4\"><span class=\"heading_side\">Calculator Inputs</span><span class=\"set_n\">Set #4</span><ul class=\"given\"><li><span class=\"input\">m₁</span> Mass of object 1</li><li><span class=\"input\">v₁</span> Final velocity of object 1</li><li><span class=\"input\">m₂</span> Mass of object 2</li><li><span class=\"input\">v₂</span> Final velocity of object 2</li><li><span class=\"input\">u₁</span> Initial velocity of object 1</li></ul><p>For these given inputs,</p><span class=\"heading_side\">Outputs, Formulas</span><p>We calculate the following outputs:</p><ul class=\"find\"><li><span class=\"output_label\">Initial velocity of object 2</span><span class=\"output\">\\(u₂ = \\)\\( \\frac{ m_1 \\cdot v_1 + m_2 \\cdot v_2 - m_1 \\cdot u_1 }{ m_2 } \\)</span></li></ul></div></div><br><br><h2>Examples</h2>\n <div id=\"example_1\" class=\"example\">\n <span class=\"example_n\">1</span>\n <h3 class=\"question\">\n Consider that there is a head on collision of two objects. The collision is perfectly elastic. The two objects are having a mass of <strong>10 kg</strong> and <strong>8 kg</strong> with initial velocities of <strong>20 m/s</strong> and <strong>4 m/s</strong> respectively.<br>What is the final velocity of first object when the final velocity of second object is <strong>16 m/s</strong>?\n </h3>\n <h4 class=\"answer\">Answer</h4>\n <p>Given:</p>\n <ul><li>Mass of object 1, m₁ = 10 kg</li><li>Initial velocity of object 1, u₁ = 20 m/s</li><li>Mass of object 2, m₂ = 8 kg</li><li>Initial velocity of object 2, u₂ = 4 m/s</li><li>Final velocity of object 2, v₂ = 16 m/s</li></ul>\n <p>Calculating final velocity of object 1 (v₁)...</p><p class=\"step\">\\( v₁ = \\) \\( \\frac{ m₁ \\times u₁ + m₂ \\times u₂ - m₂ \\times v₂ }{ m₁ } \\)</p><p class=\"step\">\\( v₁ = \\) \\( \\frac{10\\times20+8\\times4-8\\times16}{10} \\)</p><p class=\"step answer\">\\( v₁ = \\) 10.4 m/s</p>\n </div>\n \n <div id=\"example_2\" class=\"example\">\n <span class=\"example_n\">2</span>\n <h3 class=\"question\">\n Consider that there is a head on collision of two objects. The collision is perfectly elastic. The two objects are having a mass of <strong>1 kg</strong> and <strong>1 kg</strong> with initial velocities of <strong>10 m/s</strong> and <strong>5 m/s</strong> respectively.<br>What is the final velocity of first object when the final velocity of second object is <strong>10 m/s</strong>?\n </h3>\n <h4 class=\"answer\">Answer</h4>\n <p>Given:</p>\n <ul><li>Mass of object 1, m₁ = 1 kg</li><li>Initial velocity of object 1, u₁ = 10 m/s</li><li>Mass of object 2, m₂ = 1 kg</li><li>Initial velocity of object 2, u₂ = 5 m/s</li><li>Final velocity of object 2, v₂ = 10 m/s</li></ul>\n <p>Calculating final velocity of object 1 (v₁)...</p><p class=\"step\">\\( v₁ = \\) \\( \\frac{ m₁ \\times u₁ + m₂ \\times u₂ - m₂ \\times v₂ }{ m₁ } \\)</p><p class=\"step\">\\( v₁ = \\) \\( \\frac{1\\times10+1\\times5-1\\times10}{1} \\)</p><p class=\"step answer\">\\( v₁ = \\) 5 m/s</p>\n </div>\n "
}
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