What is Maximum Height in Projectile Motion? | Simple Explanation for Class 10

S6-SA2-0126

What is the Maximum Height in Projectile Motion (Trigonometry link)?

Grade Level:

Class 10

AI/ML, Physics, Biotechnology, Space Technology, Chemistry, Engineering, Medicine

Definition

What is it?

The Maximum Height in projectile motion is the highest vertical distance an object reaches from its starting point before it starts falling back down. It is determined by the object's initial speed and the angle at which it is launched, with trigonometry helping us calculate this height.

Simple Example

Quick Example

Imagine you hit a cricket ball. It goes up, reaches a certain height, and then comes down. The highest point the ball reaches in the air before descending is its maximum height. If you hit it very hard and at a good angle, it will go higher.

Worked Example

Step-by-Step

Let's find the maximum height for a stone thrown with an initial velocity of 20 m/s at an angle of 30 degrees above the horizontal. (Assume g = 10 m/s^2)

Step 1: Write down the given values.
Initial velocity (u) = 20 m/s
Angle of projection (theta) = 30 degrees
Acceleration due to gravity (g) = 10 m/s^2

---Step 2: Recall the formula for maximum height (H).
H = (u^2 * sin^2(theta)) / (2g)

---Step 3: Find the value of sin(theta).
sin(30 degrees) = 0.5

---Step 4: Calculate sin^2(theta).
sin^2(30 degrees) = (0.5)^2 = 0.25

---Step 5: Substitute the values into the formula.
H = (20^2 * 0.25) / (2 * 10)
H = (400 * 0.25) / 20

---Step 6: Perform the multiplication and division.
H = 100 / 20
H = 5 meters

---Answer: The maximum height reached by the stone is 5 meters.

Why It Matters

Understanding maximum height is crucial in fields like Space Technology for rocket launches and Engineering for designing bridges. It helps engineers and scientists predict how high objects will go, which is vital for safe and efficient operations, even helping design better sports equipment.

Common Mistakes

MISTAKE: Using sin(theta) instead of sin^2(theta) in the formula. | CORRECTION: Remember the formula for maximum height uses the square of the sine of the angle, i.e., sin(theta) multiplied by itself.

MISTAKE: Forgetting to divide by 2g in the formula. | CORRECTION: The denominator in the maximum height formula is always 2 multiplied by the acceleration due to gravity (g).

MISTAKE: Using the angle in radians instead of degrees when calculating sin(theta) with a calculator set to degrees. | CORRECTION: Always ensure your calculator is in 'DEG' mode when using angles in degrees for trigonometric functions, or convert degrees to radians if the formula requires radians.

Practice Questions

Try It Yourself

QUESTION: A football is kicked with an initial velocity of 15 m/s at an angle of 45 degrees. What is its maximum height? (Take g = 9.8 m/s^2) | ANSWER: Approximately 5.74 meters

QUESTION: If a projectile reaches a maximum height of 10 meters when launched at 30 degrees, what was its initial velocity? (Take g = 10 m/s^2) | ANSWER: Approximately 28.28 m/s

QUESTION: A javelin is thrown with an initial speed of 25 m/s and reaches a maximum height of 15 meters. At what angle was it thrown? (Take g = 9.8 m/s^2) | ANSWER: Approximately 58.7 degrees

MCQ

Quick Quiz

Which of these factors does NOT directly affect the maximum height reached by a projectile?

Initial velocity

Angle of projection

Mass of the projectile

Acceleration due to gravity

The Correct Answer Is:

The formula for maximum height does not include the mass of the projectile. It depends only on initial velocity, angle, and gravity.

Real World Connection

In the Real World

ISRO scientists use these calculations to plan rocket trajectories, ensuring satellites reach the correct altitude. Even in daily life, when a street vendor throws a vada pav to a customer, the maximum height it reaches before falling is a small example of projectile motion.

Key Vocabulary

Key Terms

PROJECTILE: An object thrown into the air that is only affected by gravity after launch | INITIAL VELOCITY: The speed and direction an object has at the very beginning of its motion | ANGLE OF PROJECTION: The angle at which an object is launched relative to the horizontal | ACCELERATION DUE TO GRAVITY (g): The constant acceleration objects experience due to Earth's gravity, approx. 9.8 m/s^2 | TRIGONOMETRY: A branch of mathematics dealing with the relationships between the sides and angles of triangles

What's Next

What to Learn Next

Next, you can explore the 'Range of Projectile Motion' and 'Time of Flight'. These concepts build on maximum height and will help you fully understand how objects move through the air, completing your knowledge of projectile motion.