Enter the initial velocity, angle of projection, and height.
How to use this Projectile Motion Calculator 🤔
There are input fields for Initial velocity \((u)\), Angle of lanuch \((θ)\), Initial height \((h)\), Acceleration due to gravity \((g)\), Range \((R)\), Maximum Height \((H)\), and Time of Flight \((T)\). Enter the initial velocity, angle of projection, and height..
The calculator uses the 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 Projectile Motion?
Projectile motion refers to the motion of an object that is launched into the air and moves under the influence of gravity. The object follows a curved path known as a parabola. Key aspects of projectile motion include the horizontal range, maximum height, and total time of flight.
Calculating Parameters of Projectile Motion
Projectile motion involves several key formulas that help determine different aspects of the motion:
Horizontal Range (R): The total distance traveled horizontally.
\( R = u \cdot \cos( θ ) \cdot \left( \frac{ u \cdot \sin( θ ) + \sqrt{ \left( u \cdot \sin( θ ) \right)^2 + 2 \cdot g \cdot h } }{ g } \right) \)
Maximum Height (H): The highest point reached by the object during its motion.
\( H = h + \frac{ u^2 \cdot \sin^2( θ ) }{ 2 \cdot g } \)
Total Time of Flight (T): The total time the object remains in the air.
\( T = \frac{ u \cdot \sin( θ ) + \sqrt{ \left( u \cdot \sin( θ ) \right)^2 + 2 \cdot g \cdot h } }{ g } \)
In these formulas, the parameters are defined as follows:
u: The initial velocity, or the speed at which the object is launched.
θ: The angle of projection, or the angle at which the object is launched relative to the horizontal.
g: The acceleration due to gravity, typically taken as 9.8 m/s².
h: The initial height from which the object is launched.
Input and Output Combinations of the Projectile Motion Calculator
Here are the input combinations available for the calculator, along with the formulas it uses to calculate the corresponding outputs.
Calculator InputsSet #1
u Initial velocity
θ Angle of lanuch
h Initial height
For these given inputs,
Outputs, Formulas
We calculate the following outputs:
Range\(R = \)\( u \cdot \cos( θ ) \cdot \left( \frac{ u \cdot \sin( θ ) + \sqrt{ \left( u \cdot \sin( θ ) \right)^2 + 2 \cdot g \cdot h } }{ g } \right) \)
Maximum Height\(H = \)\( h + \frac{ u^2 \cdot \sin^2( θ ) }{ 2 \cdot g } \)
Time of Flight\(T = \)\( \frac{ u \cdot \sin( θ ) + \sqrt{ \left( u \cdot \sin( θ ) \right)^2 + 2 \cdot g \cdot h } }{ g } \)
Consider that an athlete has thrown a Javelin at an angle of 45 ° with an initial velocity of 30 m/s. The javelin is released from hand at a height of 1.2 m from the ground. Calculate the horizontal distance travelled by the javelin, the maximum height it travelled from the ground, and the time of flight.
Answer
Given:
initial velocity of object, u = 30 m/s
angle of launch, θ = 45 °
initial height from which the object is launched, h = 1.2 m
Calculating range (R)...
\( R = \) \( u \cdot \cos( θ ) \cdot \left( \frac{ u \cdot \sin( θ ) + \sqrt{ \left( u \cdot \sin( θ ) \right)^2 + 2 \cdot g \cdot h } }{ g } \right) \)
Consider that an athlete has thrown a Javelin at an angle of 90 ° with an initial velocity of 10 m/s. The javelin is released from hand at a height of 1 m from the ground. Calculate the horizontal distance travelled by the javelin, the maximum height it travelled from the ground, and the time of flight.
Answer
Given:
initial velocity of object, u = 10 m/s
angle of launch, θ = 90 °
initial height from which the object is launched, h = 1 m
Calculating range (R)...
\( R = \) \( u \cdot \cos( θ ) \cdot \left( \frac{ u \cdot \sin( θ ) + \sqrt{ \left( u \cdot \sin( θ ) \right)^2 + 2 \cdot g \cdot h } }{ g } \right) \)
Consider that an athlete has thrown a Javelin at an angle of 45 ° with an initial velocity of 30 m/s. The javelin is released from hand at a height of 1.2 m from the ground. Calculate the horizontal distance travelled by the javelin, the maximum height it travelled from the ground, and the time of flight.
Answer
Given:
initial velocity of object, u = 30 m/s
angle of launch, θ = 45 °
initial height from which the object is launched, h = 1.2 m
Calculating range (R)...
\( R = \) \( u \cdot \cos( θ ) \cdot \left( \frac{ u \cdot \sin( θ ) + \sqrt{ \left( u \cdot \sin( θ ) \right)^2 + 2 \cdot g \cdot h } }{ g } \right) \)
Consider that an athlete has thrown a Javelin at an angle of 90 ° with an initial velocity of 10 m/s. The javelin is released from hand at a height of 1 m from the ground. Calculate the horizontal distance travelled by the javelin, the maximum height it travelled from the ground, and the time of flight.
Answer
Given:
initial velocity of object, u = 10 m/s
angle of launch, θ = 90 °
initial height from which the object is launched, h = 1 m
Calculating range (R)...
\( R = \) \( u \cdot \cos( θ ) \cdot \left( \frac{ u \cdot \sin( θ ) + \sqrt{ \left( u \cdot \sin( θ ) \right)^2 + 2 \cdot g \cdot h } }{ g } \right) \)
{
"topic": "projectile-motion",
"input_types": [
"float",
"float",
"float",
"float",
"float",
"float",
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],
"input_labels": [
"u",
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"H",
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],
"input_descriptions": [
"Initial velocity",
"Angle of lanuch",
"Initial height",
"Acceleration due to gravity",
"Range",
"Maximum Height",
"Time of Flight"
],
"input_descriptions_long": [
"initial velocity of object",
"angle of launch",
"initial height from which the object is launched",
"Acceleration due to gravity",
"range or distance travelled by the object",
"maximum height travelled by object",
"time of flight"
],
"input_units": [
"m/s",
"°",
"m",
"m/s²",
"m",
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],
"input_values": [
"",
"",
"",
"9.81",
"",
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""
],
"input_pre_msg": "Enter the initial velocity, angle of projection, and height.",
"type": "Calculate",
"title": "Projectile Motion Calculator",
"description": "Use our Projectile Motion Calculator to compute the range, maximum height, and time of flight for a projectile launched with a given initial velocity and angle. Ideal for physics students and educators.",
"category": "Kinematics",
"template": "physics",
"precision": 10,
"formulas": [
{
"parameters": [
"u",
"θ",
"h"
],
"outputs": [
{
"label": "R",
"formula": " u * Math.cos( θ * Math.PI / 180 ) *(( u * Math.sin( θ * Math.PI / 180 ) + Math.sqrt( Math.pow( u * Math.sin( θ * Math.PI / 180 ), 2) + 2* g * h ) ) / g )",
"formula_mathjax": "\\( u \\cdot \\cos( θ ) \\cdot \\left( \\frac{ u \\cdot \\sin( θ ) + \\sqrt{ \\left( u \\cdot \\sin( θ ) \\right)^2 + 2 \\cdot g \\cdot h } }{ g } \\right) \\)",
"formula_mathjax_sub": "\\( u \\times \\cos( θ °) \\times \\left( \\frac{ u \\times \\sin( θ °) + \\sqrt{ \\left( u \\times \\sin( θ °) \\right)^2 + 2 \\times g \\times h } }{ g } \\right) \\)"
},
{
"label": "H",
"formula": " h + ( u * u * Math.pow(Math.sin( θ * Math.PI / 180 ), 2)) / (2 * g )",
"formula_mathjax": "\\( h + \\frac{ u^2 \\cdot \\sin^2( θ ) }{ 2 \\cdot g } \\)",
"formula_mathjax_sub": "\\( h + \\frac{ u^2 \\times \\sin^2( θ °) }{ 2 \\times g } \\)"
},
{
"label": "T",
"formula": "( u * Math.sin( θ * Math.PI / 180 ) + Math.sqrt( Math.pow( u * Math.sin( θ * Math.PI / 180 ), 2) + 2* g * h ) ) / g ",
"formula_mathjax": "\\( \\frac{ u \\cdot \\sin( θ ) + \\sqrt{ \\left( u \\cdot \\sin( θ ) \\right)^2 + 2 \\cdot g \\cdot h } }{ g } \\)",
"formula_mathjax_sub": "\\( \\frac{ u \\times \\sin( θ °) + \\sqrt{ \\left( u \\times \\sin( θ °) \\right)^2 + 2 \\times g \\times h } }{ g } \\)"
}
],
"example_heading": "Consider that an athlete has thrown a Javelin at an angle of _θ_ with an initial velocity of _u_. The javelin is released from hand at a height of _h_ from the ground.<br> Calculate the horizontal distance travelled by the javelin, the maximum height it travelled from the ground, and the time of flight.",
"examples": [
{
"u": 30,
"θ": 45,
"h": 1.2,
"g": 9.81,
"R": "92.9278",
"H": "24.1358",
"T": "4.3807"
},
{
"u": 10,
"θ": 90,
"h": 1,
"g": 9.81,
"R": "0",
"H": "6.0968",
"T": "2.1343"
}
]
}
],
"content": "<h2>What is Projectile Motion?</h2><p>Projectile motion refers to the motion of an object that is launched into the air and moves under the influence of gravity. The object follows a curved path known as a parabola. Key aspects of projectile motion include the horizontal range, maximum height, and total time of flight.</p><h2>Calculating Parameters of Projectile Motion</h2><p>Projectile motion involves several key formulas that help determine different aspects of the motion:</p><ul><li><strong>Horizontal Range (R)</strong>: The total distance traveled horizontally.</li><p class=\"formula_highlite\">\\( R = u \\cdot \\cos( θ ) \\cdot \\left( \\frac{ u \\cdot \\sin( θ ) + \\sqrt{ \\left( u \\cdot \\sin( θ ) \\right)^2 + 2 \\cdot g \\cdot h } }{ g } \\right) \\)</p><li><strong>Maximum Height (H)</strong>: The highest point reached by the object during its motion.</li><p class=\"formula_highlite\">\\( H = h + \\frac{ u^2 \\cdot \\sin^2( θ ) }{ 2 \\cdot g } \\)</p><li><strong>Total Time of Flight (T)</strong>: The total time the object remains in the air.</li><p class=\"formula_highlite\">\\( T = \\frac{ u \\cdot \\sin( θ ) + \\sqrt{ \\left( u \\cdot \\sin( θ ) \\right)^2 + 2 \\cdot g \\cdot h } }{ g } \\)</p></ul><p>In these formulas, the parameters are defined as follows:</p><ul><li><strong>u</strong>: The initial velocity, or the speed at which the object is launched.</li><li><strong>θ</strong>: The angle of projection, or the angle at which the object is launched relative to the horizontal.</li><li><strong>g</strong>: The acceleration due to gravity, typically taken as 9.8 m/s².</li><li><strong>h</strong>: The initial height from which the object is launched.</li></ul><br><h2>Input and Output Combinations of the Projectile Motion 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\">u</span> Initial velocity</li><li><span class=\"input\">θ</span> Angle of lanuch</li><li><span class=\"input\">h</span> Initial height</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\">Range</span><span class=\"output\">\\(R = \\)\\( u \\cdot \\cos( θ ) \\cdot \\left( \\frac{ u \\cdot \\sin( θ ) + \\sqrt{ \\left( u \\cdot \\sin( θ ) \\right)^2 + 2 \\cdot g \\cdot h } }{ g } \\right) \\)</span></li><li><span class=\"output_label\">Maximum Height</span><span class=\"output\">\\(H = \\)\\( h + \\frac{ u^2 \\cdot \\sin^2( θ ) }{ 2 \\cdot g } \\)</span></li><li><span class=\"output_label\">Time of Flight</span><span class=\"output\">\\(T = \\)\\( \\frac{ u \\cdot \\sin( θ ) + \\sqrt{ \\left( u \\cdot \\sin( θ ) \\right)^2 + 2 \\cdot g \\cdot h } }{ g } \\)</span></li></ul><a class=\"button\" href=\"#example_1\">2 Examples</a></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 an athlete has thrown a Javelin at an angle of <strong>45 °</strong> with an initial velocity of <strong>30 m/s</strong>. The javelin is released from hand at a height of <strong>1.2 m</strong> from the ground.<br> Calculate the horizontal distance travelled by the javelin, the maximum height it travelled from the ground, and the time of flight.\n </h3>\n <h4 class=\"answer\">Answer</h4>\n <p>Given:</p>\n <ul><li>initial velocity of object, u = 30 m/s</li><li>angle of launch, θ = 45 °</li><li>initial height from which the object is launched, h = 1.2 m</li></ul>\n <p>Calculating range (R)...</p><p class=\"step\">\\( R = \\) \\( u \\cdot \\cos( θ ) \\cdot \\left( \\frac{ u \\cdot \\sin( θ ) + \\sqrt{ \\left( u \\cdot \\sin( θ ) \\right)^2 + 2 \\cdot g \\cdot h } }{ g } \\right) \\)</p><p class=\"step\">\\( R = \\) \\(30\\times \\cos(45°) \\times \\left( \\frac{30\\times \\sin(45°) + \\sqrt{ \\left(30\\times \\sin(45°) \\right)^2 + 2 \\times9.81\\times1.2} }{9.81} \\right) \\)</p><p class=\"step answer\">\\( R = \\) 92.9278 m</p><p>Calculating maximum height (H)...</p><p class=\"step\">\\( H = \\) \\( h + \\frac{ u^2 \\cdot \\sin^2( θ ) }{ 2 \\cdot g } \\)</p><p class=\"step\">\\( H = \\) \\(1.2+ \\frac{ u^2 \\times \\sin^2(45°) }{ 2 \\times9.81} \\)</p><p class=\"step answer\">\\( H = \\) 24.1358 m</p><p>Calculating time of flight (T)...</p><p class=\"step\">\\( T = \\) \\( \\frac{ u \\cdot \\sin( θ ) + \\sqrt{ \\left( u \\cdot \\sin( θ ) \\right)^2 + 2 \\cdot g \\cdot h } }{ g } \\)</p><p class=\"step\">\\( T = \\) \\( \\frac{30\\times \\sin(45°) + \\sqrt{ \\left(30\\times \\sin(45°) \\right)^2 + 2 \\times9.81\\times1.2} }{9.81} \\)</p><p class=\"step answer\">\\( T = \\) 4.3807 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 an athlete has thrown a Javelin at an angle of <strong>90 °</strong> with an initial velocity of <strong>10 m/s</strong>. The javelin is released from hand at a height of <strong>1 m</strong> from the ground.<br> Calculate the horizontal distance travelled by the javelin, the maximum height it travelled from the ground, and the time of flight.\n </h3>\n <h4 class=\"answer\">Answer</h4>\n <p>Given:</p>\n <ul><li>initial velocity of object, u = 10 m/s</li><li>angle of launch, θ = 90 °</li><li>initial height from which the object is launched, h = 1 m</li></ul>\n <p>Calculating range (R)...</p><p class=\"step\">\\( R = \\) \\( u \\cdot \\cos( θ ) \\cdot \\left( \\frac{ u \\cdot \\sin( θ ) + \\sqrt{ \\left( u \\cdot \\sin( θ ) \\right)^2 + 2 \\cdot g \\cdot h } }{ g } \\right) \\)</p><p class=\"step\">\\( R = \\) \\(10\\times \\cos(90°) \\times \\left( \\frac{10\\times \\sin(90°) + \\sqrt{ \\left(10\\times \\sin(90°) \\right)^2 + 2 \\times9.81\\times1} }{9.81} \\right) \\)</p><p class=\"step answer\">\\( R = \\) 0 m</p><p>Calculating maximum height (H)...</p><p class=\"step\">\\( H = \\) \\( h + \\frac{ u^2 \\cdot \\sin^2( θ ) }{ 2 \\cdot g } \\)</p><p class=\"step\">\\( H = \\) \\(1+ \\frac{ u^2 \\times \\sin^2(90°) }{ 2 \\times9.81} \\)</p><p class=\"step answer\">\\( H = \\) 6.0968 m</p><p>Calculating time of flight (T)...</p><p class=\"step\">\\( T = \\) \\( \\frac{ u \\cdot \\sin( θ ) + \\sqrt{ \\left( u \\cdot \\sin( θ ) \\right)^2 + 2 \\cdot g \\cdot h } }{ g } \\)</p><p class=\"step\">\\( T = \\) \\( \\frac{10\\times \\sin(90°) + \\sqrt{ \\left(10\\times \\sin(90°) \\right)^2 + 2 \\times9.81\\times1} }{9.81} \\)</p><p class=\"step answer\">\\( T = \\) 2.1343 s</p>\n </div>\n <br><h2>Input and Output Combinations of the Projectile Motion 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\">u</span> Initial velocity</li><li><span class=\"input\">θ</span> Angle of lanuch</li><li><span class=\"input\">h</span> Initial height</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\">Range</span><span class=\"output\">\\(R = \\)\\( u \\cdot \\cos( θ ) \\cdot \\left( \\frac{ u \\cdot \\sin( θ ) + \\sqrt{ \\left( u \\cdot \\sin( θ ) \\right)^2 + 2 \\cdot g \\cdot h } }{ g } \\right) \\)</span></li><li><span class=\"output_label\">Maximum Height</span><span class=\"output\">\\(H = \\)\\( h + \\frac{ u^2 \\cdot \\sin^2( θ ) }{ 2 \\cdot g } \\)</span></li><li><span class=\"output_label\">Time of Flight</span><span class=\"output\">\\(T = \\)\\( \\frac{ u \\cdot \\sin( θ ) + \\sqrt{ \\left( u \\cdot \\sin( θ ) \\right)^2 + 2 \\cdot g \\cdot h } }{ g } \\)</span></li></ul><a class=\"button\" href=\"#example_1\">2 Examples</a></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 an athlete has thrown a Javelin at an angle of <strong>45 °</strong> with an initial velocity of <strong>30 m/s</strong>. The javelin is released from hand at a height of <strong>1.2 m</strong> from the ground.<br> Calculate the horizontal distance travelled by the javelin, the maximum height it travelled from the ground, and the time of flight.\n </h3>\n <h4 class=\"answer\">Answer</h4>\n <p>Given:</p>\n <ul><li>initial velocity of object, u = 30 m/s</li><li>angle of launch, θ = 45 °</li><li>initial height from which the object is launched, h = 1.2 m</li></ul>\n <p>Calculating range (R)...</p><p class=\"step\">\\( R = \\) \\( u \\cdot \\cos( θ ) \\cdot \\left( \\frac{ u \\cdot \\sin( θ ) + \\sqrt{ \\left( u \\cdot \\sin( θ ) \\right)^2 + 2 \\cdot g \\cdot h } }{ g } \\right) \\)</p><p class=\"step\">\\( R = \\) \\(30\\times \\cos(45°) \\times \\left( \\frac{30\\times \\sin(45°) + \\sqrt{ \\left(30\\times \\sin(45°) \\right)^2 + 2 \\times9.81\\times1.2} }{9.81} \\right) \\)</p><p class=\"step answer\">\\( R = \\) 92.9278 m</p><p>Calculating maximum height (H)...</p><p class=\"step\">\\( H = \\) \\( h + \\frac{ u^2 \\cdot \\sin^2( θ ) }{ 2 \\cdot g } \\)</p><p class=\"step\">\\( H = \\) \\(1.2+ \\frac{ u^2 \\times \\sin^2(45°) }{ 2 \\times9.81} \\)</p><p class=\"step answer\">\\( H = \\) 24.1358 m</p><p>Calculating time of flight (T)...</p><p class=\"step\">\\( T = \\) \\( \\frac{ u \\cdot \\sin( θ ) + \\sqrt{ \\left( u \\cdot \\sin( θ ) \\right)^2 + 2 \\cdot g \\cdot h } }{ g } \\)</p><p class=\"step\">\\( T = \\) \\( \\frac{30\\times \\sin(45°) + \\sqrt{ \\left(30\\times \\sin(45°) \\right)^2 + 2 \\times9.81\\times1.2} }{9.81} \\)</p><p class=\"step answer\">\\( T = \\) 4.3807 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 an athlete has thrown a Javelin at an angle of <strong>90 °</strong> with an initial velocity of <strong>10 m/s</strong>. The javelin is released from hand at a height of <strong>1 m</strong> from the ground.<br> Calculate the horizontal distance travelled by the javelin, the maximum height it travelled from the ground, and the time of flight.\n </h3>\n <h4 class=\"answer\">Answer</h4>\n <p>Given:</p>\n <ul><li>initial velocity of object, u = 10 m/s</li><li>angle of launch, θ = 90 °</li><li>initial height from which the object is launched, h = 1 m</li></ul>\n <p>Calculating range (R)...</p><p class=\"step\">\\( R = \\) \\( u \\cdot \\cos( θ ) \\cdot \\left( \\frac{ u \\cdot \\sin( θ ) + \\sqrt{ \\left( u \\cdot \\sin( θ ) \\right)^2 + 2 \\cdot g \\cdot h } }{ g } \\right) \\)</p><p class=\"step\">\\( R = \\) \\(10\\times \\cos(90°) \\times \\left( \\frac{10\\times \\sin(90°) + \\sqrt{ \\left(10\\times \\sin(90°) \\right)^2 + 2 \\times9.81\\times1} }{9.81} \\right) \\)</p><p class=\"step answer\">\\( R = \\) 0 m</p><p>Calculating maximum height (H)...</p><p class=\"step\">\\( H = \\) \\( h + \\frac{ u^2 \\cdot \\sin^2( θ ) }{ 2 \\cdot g } \\)</p><p class=\"step\">\\( H = \\) \\(1+ \\frac{ u^2 \\times \\sin^2(90°) }{ 2 \\times9.81} \\)</p><p class=\"step answer\">\\( H = \\) 6.0968 m</p><p>Calculating time of flight (T)...</p><p class=\"step\">\\( T = \\) \\( \\frac{ u \\cdot \\sin( θ ) + \\sqrt{ \\left( u \\cdot \\sin( θ ) \\right)^2 + 2 \\cdot g \\cdot h } }{ g } \\)</p><p class=\"step\">\\( T = \\) \\( \\frac{10\\times \\sin(90°) + \\sqrt{ \\left(10\\times \\sin(90°) \\right)^2 + 2 \\times9.81\\times1} }{9.81} \\)</p><p class=\"step answer\">\\( T = \\) 2.1343 s</p>\n </div>\n "
}
ConvertOnline