Galileo Projectile Motion Theorem Explained Step by Step

 Dr.sanjaykumar pawar 

Projectile motion diagram showing why complementary angles produce equal horizontal ranges. 

Internal Links

Introduction to Projectile Motion

Derivation of Range Formula

Maximum Height Formula in Projectile Motion

Time of Flight Explained

Motion in a Plane Class 11 Notes

Trigonometric Identities Used in Physics

Important Projectile Motion Numericals

NCERT Solutions for Class 11 Physics

JEE Projectile Motion Revision Notes

NEET Physics Important Formulas

Example 3.6 – Galileo’s Statement

Example 3.6 Galileo, in his book Two new sciences, stated that “for elevations which exceed or fall short of 45° by equal amounts, the ranges are equal”. Prove this statement.

Question

Galileo stated:

“For elevations which exceed or fall short of 45° by equal amounts, the ranges are equal.”

Prove this statement.

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Step 1: Write the Formula for Range

For a projectile projected with speed v₀ at an angle θ, the horizontal range is:

R = (v₀² sin 2θ) / g

Where:

• R = Horizontal Range
• v₀ = Initial Velocity
• g = Acceleration due to gravity
• θ = Angle of projection

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Step 2: Consider Two Angles

Take two angles:

(45° + α)

and

(45° − α)

These angles are equally above and below 45°.

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Step 3: Find 2θ for Both Angles

For the first angle:

2θ = 2(45° + α)
= 90° + 2α

For the second angle:

2θ = 2(45° − α)
= 90° − 2α

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Step 4: Write the Sine Terms

The range depends on sin 2θ.

So we get:

sin(90° + 2α)

and

sin(90° − 2α)

Using trigonometric identities:

sin(90° + x) = cos x

sin(90° − x) = cos x

Therefore:

sin(90° + 2α) = cos 2α

sin(90° − 2α) = cos 2α

Both values are equal.

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Step 5: Compare the Ranges

Since the value of sin 2θ is the same for both angles, the range formula gives the same result.

R₁ = R₂

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Final Conclusion

Hence, the ranges are equal for projection angles that exceed or fall short of 45° by the same amount.

Therefore, Galileo’s statement is proved.

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Quick Revision Point

Examples of equal ranges:

• 30° and 60°
• 40° and 50°

Because complementary angles produce equal horizontal ranges.