Two projectiles are fired with same initial speed from same point on ground at angles of (45∘−α) and(45∘+α), respectively, with the horizontal direction. The ratio of their maximum height attained is : JEE MAINS 2025 (29 JAN SHIFT 1)

Two projectiles are fired with same initial speed from same point on ground at angles of (45∘−α) and(45∘+α), respectively, with the horizontal direction. The ratio of their maximum height attained is:

A) (1 + tan α) / (1 - tan α)

B) (1 + sin 2α) / (1 - sin 2α) (Answer)

C) 1 / (1 + sin 2α)

D) 1 / (1 + sin α)

Projectile motion is an important topic in JEE Mains, and understanding it conceptually helps in solving problems quickly. Many students rely on memorizing formulas, but a deeper understanding will help tackle variations of such problems with confidence.

In this article, we will go through a step-by-step solution to a common JEE-level projectile motion problem while ensuring a logical flow that keeps readers engaged until the end.

Problem Statement

Two projectiles are launched from the same point on the ground with the same initial speed (u) but at different angles:

1. First projectile: Launched at an angle of (45° - α) with the horizontal.
2. Second projectile: Launched at an angle of (45° + α) with the horizontal.

We need to determine the ratio of their maximum heights (H1 / H2).

Understanding Maximum Height in Projectile Motion
The maximum height (H) of a projectile is the highest point it reaches before starting its descent. It occurs when the vertical component of velocity becomes zero due to gravity.

The formula for maximum height in projectile motion is:
H = (u² sin²θ) / (2g)

where:
H is the maximum height,
u is the initial velocity,
θ is the angle of projection,
g is the acceleration due to gravity.

Since both projectiles have the same initial speed (u), we apply this formula separately for each angle.

Step 1: Maximum Height for the First Projectile
For the first projectile, the angle of projection is (45° - α). Using the height formula:
H1 = (u² sin²(45° - α)) / (2g)

Step 2: Maximum Height for the Second Projectile
For the second projectile, the angle of projection is (45° + α). Applying the same formula:

H2 = (u² sin²(45° + α)) / (2g)

Step 3: Finding the Ratio of Maximum Heights
To find the ratio H₁ / H₂, we divide both equations:
H1 / H2 = sin²(45° - α) / sin²(45° + α)

At this point, many students would just substitute values, but let’s take a deeper look at why this formula works.

We use the trigonometric identity for sine of compound angles:
sin(45° ± α) = (cos α ± sin α) / √2

Squaring both sides:
sin²(45° - α) = [(cos α - sin α)²] / 2
sin²(45° + α) = [(cos α + sin α)²] / 2

Now, dividing the two:
H1 / H2 = [(cos α - sin α)²] / [(cos α + sin α)²]

Expanding both squares:
H1 / H2 = (cos²α - 2cosα sinα + sin²α) / (cos²α + 2cosα sinα + sin²α)

Since cos²α + sin²α = 1, we simplify:
H1 / H2 = (1 - 2cosα sinα) / (1 + 2cosα sinα)

Using sin 2α = 2sinα cosα,
we get:
H1 / H2 = (1 - sin 2α) / (1 + sin 2α)

Final Answer:
The ratio of the maximum heights of the two projectiles is:
(1 - sin 2α) / (1 + sin 2α)

Key Takeaways for JEE Mains

The formula for maximum height in projectile motion is H = (u² sin²θ) / (2g).

When two projectiles are launched with the same speed but at angles (45° - α) and (45° + α), their height ratio is:
(1 - sin 2α) / (1 + sin 2α)

Understanding trigonometric identities is key to simplifying these problems quickly in JEE Mains.

Common Mistakes Students Make

1. Forgetting to square the sine function – Many students incorrectly write sin(45° - α) / sin(45° + α) instead of sin²(45° - α) / sin²(45° + α).

2. Not using the identity for sin(45° ± α) – Without this, simplifying the expression becomes difficult.

3. Ignoring why we divide H₁ by H₂ – Understanding why we compare heights makes solving similar problems easier.

Conclusion

Understanding projectile motion conceptually instead of just memorizing formulas can make JEE Mains problems much easier. The key is to analyze why we use each step rather than just applying formulas blindly.

Next time you see a similar problem, try solving it step by step without referring to the formula directly. If you have understood every step here, you should be able to solve it easily in the exam!

Let me know in the comments if you found this explanation helpful.

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