A Man Can Throw A Stone To A Maximum Distance Of 40 M. What Is The Maximum Height To Which He Can Throw It?

Jun 1, 2025 by ADMIN 108 views

A Man Can Throw a Stone to a Maximum Distance of 40 m: What is the Maximum Height to Which He Can Throw It?

Introduction

When a man throws a stone, it follows a curved trajectory under the influence of gravity. The maximum distance the stone can travel is determined by the initial velocity and the angle of projection. However, the maximum height to which the stone can be thrown is a different story. In this article, we will explore the relationship between the maximum distance and the maximum height of a projectile motion.

Understanding Projectile Motion

Projectile motion is a type of motion where an object moves under the influence of gravity. The motion can be broken down into two components: horizontal and vertical. The horizontal component is independent of gravity, while the vertical component is affected by gravity. When a stone is thrown, it follows a parabolic path, with the horizontal component determining the range and the vertical component determining the maximum height.

The Relationship Between Maximum Distance and Maximum Height

The maximum distance a stone can travel is given by the equation:

R = (v^2 * sin(2θ)) / g

where R is the range, v is the initial velocity, θ is the angle of projection, and g is the acceleration due to gravity.

The maximum height to which the stone can be thrown is given by the equation:

H = (v^2 * sin^2(θ)) / (2 * g)

where H is the maximum height.

Deriving the Relationship Between Maximum Distance and Maximum Height

To derive the relationship between the maximum distance and the maximum height, we can start by considering the equation for the range:

R = (v^2 * sin(2θ)) / g

We can rewrite this equation as:

R = (v^2 * 2 * sin(θ) * cos(θ)) / g

Now, we can use the trigonometric identity:

sin(2θ) = 2 * sin(θ) * cos(θ)

Substituting this into the equation for the range, we get:

R = (v^2 * sin(2θ)) / g

Now, we can divide the equation for the range by the equation for the maximum height:

R / H = ((v^2 * sin(2θ)) / g) / ((v^2 * sin^2(θ)) / (2 * g))

Simplifying this equation, we get:

R / H = 2 * cos(θ)

Maximum Height to Maximum Distance Ratio

The ratio of the maximum height to the maximum distance is given by:

H / R = 1 / (2 * cos(θ))

This equation shows that the ratio of the maximum height to the maximum distance is inversely proportional to the cosine of the angle of projection.

Maximum Height to Maximum Distance Ratio for a Given Angle

If the angle of projection is 45 degrees, the ratio of the maximum height to the maximum distance is:

H / R = 1 / (2 * cos(45°))

Using the value of cos(45°) = 1/√2, we get:

H / R = 1 / (2 * 1/√2)

Simplifying this equation, we get:

H / R = √2 / 2

Maximum Height to Maximum Distance Ratio for a Given Angle of 60 Degrees

If angle of projection is 60 degrees, the ratio of the maximum height to the maximum distance is:

H / R = 1 / (2 * cos(60°))

Using the value of cos(60°) = 1/2, we get:

H / R = 1 / (2 * 1/2)

Simplifying this equation, we get:

H / R = 1

Maximum Height to Maximum Distance Ratio for a Given Angle of 90 Degrees

If the angle of projection is 90 degrees, the ratio of the maximum height to the maximum distance is:

H / R = 1 / (2 * cos(90°))

Using the value of cos(90°) = 0, we get:

H / R = 1 / (2 * 0)

This equation is undefined, which means that the ratio of the maximum height to the maximum distance is not defined for an angle of 90 degrees.

Conclusion

In conclusion, the maximum height to which a stone can be thrown is related to the maximum distance it can travel. The ratio of the maximum height to the maximum distance is inversely proportional to the cosine of the angle of projection. By understanding this relationship, we can determine the maximum height to which a stone can be thrown for a given angle of projection.

References

Halliday, D., Resnick, R., & Walker, J. (2013). Fundamentals of Physics. John Wiley & Sons.
Serway, R. A., & Jewett, J. W. (2018). Physics for Scientists and Engineers. Cengage Learning.

Introduction

In our previous article, we explored the relationship between the maximum distance and the maximum height of a projectile motion. We derived the equation for the ratio of the maximum height to the maximum distance and discussed its implications. In this article, we will answer some frequently asked questions related to this topic.

Q&A

Q: What is the maximum height to which a stone can be thrown if it is thrown at an angle of 45 degrees?

A: To find the maximum height, we need to use the equation:

H = (v^2 * sin^2(θ)) / (2 * g)

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

For an angle of 45 degrees, we have:

H = (v^2 * sin^2(45°)) / (2 * g)

Using the value of sin(45°) = 1/√2, we get:

H = (v^2 * (1/√2)^2) / (2 * g)

Simplifying this equation, we get:

H = (v^2 * 1/2) / (2 * g)

H = (v^2) / (4 * g)

Q: What is the maximum height to which a stone can be thrown if it is thrown at an angle of 60 degrees?

A: To find the maximum height, we need to use the equation:

H = (v^2 * sin^2(θ)) / (2 * g)

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

For an angle of 60 degrees, we have:

H = (v^2 * sin^2(60°)) / (2 * g)

Using the value of sin(60°) = √3/2, we get:

H = (v^2 * (√3/2)^2) / (2 * g)

Simplifying this equation, we get:

H = (v^2 * 3/4) / (2 * g)

H = (3 * v^2) / (8 * g)

Q: What is the maximum height to which a stone can be thrown if it is thrown at an angle of 90 degrees?

A: To find the maximum height, we need to use the equation:

H = (v^2 * sin^2(θ)) / (2 * g)

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

For an angle of 90 degrees, we have:

H = (v^2 * sin^2(90°)) / (2 * g)

Using the value of sin(90°) = 1, we get:

H = (v^2 * 1^2) / (2 * g)

Simplifying this equation, we get:

H = (v^2) / (2 * g)

Q: What is the relationship between the maximum height and the maximum distance?

A: The ratio of the maximum height to the maximum distance is given:

H / R = 1 / (2 * cos(θ))

where H is the maximum height, R is the maximum distance, and θ is the angle of projection.

This equation shows that the ratio of the maximum height to the maximum distance is inversely proportional to the cosine of the angle of projection.

Q: What is the maximum height to which a stone can be thrown if it is thrown at an angle of 45 degrees and the maximum distance is 40 m?

A: To find the maximum height, we need to use the equation:

H = (v^2 * sin^2(θ)) / (2 * g)

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

For an angle of 45 degrees, we have:

H = (v^2 * sin^2(45°)) / (2 * g)

Using the value of sin(45°) = 1/√2, we get:

H = (v^2 * (1/√2)^2) / (2 * g)

Simplifying this equation, we get:

H = (v^2 * 1/2) / (2 * g)

H = (v^2) / (4 * g)

We are given that the maximum distance is 40 m. Using the equation:

R = (v^2 * sin(2θ)) / g

where R is the maximum distance, v is the initial velocity, θ is the angle of projection, and g is the acceleration due to gravity.

For an angle of 45 degrees, we have:

R = (v^2 * sin(2 * 45°)) / g

Using the value of sin(90°) = 1, we get:

R = (v^2 * 1) / g

Simplifying this equation, we get:

R = (v^2) / g

We are given that the maximum distance is 40 m. Therefore:

(v^2) / g = 40

(v^2) = 40 * g

Substituting this value into the equation for the maximum height, we get:

H = ((40 * g) / g) / 4

H = 40 / 4

H = 10

Therefore, the maximum height to which the stone can be thrown is 10 m.

Conclusion

In this article, we answered some frequently asked questions related to the maximum height to which a stone can be thrown. We derived the equations for the maximum height and the maximum distance and discussed their implications. We also provided examples to illustrate the relationship between the maximum height and the maximum distance.

References

Halliday, D., Resnick, R., & Walker, J. (2013). Fundamentals of Physics. John Wiley & Sons.
Serway, R. A., & Jewett, J. W. (2018). Physics for Scientists and Engineers. Cengage Learning.

A Man Can Throw A Stone To A Maximum Distance Of 40 M. What Is The Maximum Height To Which He Can Throw It?

Introduction

Understanding Projectile Motion

The Relationship Between Maximum Distance and Maximum Height

Deriving the Relationship Between Maximum Distance and Maximum Height

Maximum Height to Maximum Distance Ratio

Maximum Height to Maximum Distance Ratio for a Given Angle

Maximum Height to Maximum Distance Ratio for a Given Angle of 60 Degrees

Maximum Height to Maximum Distance Ratio for a Given Angle of 90 Degrees

Conclusion

References

Further Reading

Introduction

Q&A

Q: What is the maximum height to which a stone can be thrown if it is thrown at an angle of 45 degrees?

Q: What is the maximum height to which a stone can be thrown if it is thrown at an angle of 60 degrees?

Q: What is the maximum height to which a stone can be thrown if it is thrown at an angle of 90 degrees?

Q: What is the relationship between the maximum height and the maximum distance?

Q: What is the maximum height to which a stone can be thrown if it is thrown at an angle of 45 degrees and the maximum distance is 40 m?

Conclusion

References

Further Reading