The maximum horizontal distance a boy can throw a ball is 52 m. Assume he can throw with the same initial speed at all angles. How high does he throw the ball when he throws it straight upward?

To solve this problem, we need to relate the maximum horizontal distance the ball can be thrown to the maximum height it reaches when thrown vertically.

### Given Information:
- Maximum horizontal distance (\( R \)) = 52 m
- Initial speed (\( v_0 \)) is the same for both horizontal and vertical throws.

### Horizontal Throw Analysis:

The maximum range \( R \) for projectile motion is given by the formula:

\[
R = \frac{v_0^2 \sin 2\theta}{g}
\]

where:
- \( v_0 \) is the initial speed,
- \( \theta \) is the angle of projection,
- \( g \) is the acceleration due to gravity (approximately \( 9.8 \, \text{m/s}^2 \)).

For maximum range, \( \theta = 45^\circ \). So, \( \sin 2\theta = \sin 90^\circ = 1 \).

Thus, the equation simplifies to:

\[
R = \frac{v_0^2}{g}
\]

Plugging in the values, we get:

\[
52 = \frac{v_0^2}{9.8}
\]

Solving for \( v_0 \):

\[
v_0^2 = 52 \times 9.8 = 509.6 \, \text{m}^2/\text{s}^2
\]
\[
v_0 = \sqrt{509.6} \approx 22.57 \, \text{m/s}
\]

### Vertical Throw Analysis:

When the ball is thrown straight up, the maximum height (\( H \)) it reaches is given by:

\[
H = \frac{v_0^2}{2g}
\]

Substitute the value of \( v_0^2 = 509.6 \):

\[
H = \frac{509.6}{2 \times 9.8}
\]
\[
H = \frac{509.6}{19.6} \approx 26 \, \text{m}
\]

Thus, the maximum height the boy can throw the ball vertically upward is approximately **26 meters**.

Would you like more details or have any questions? Here are some related questions you might find interesting:

1. How would the maximum height change if the acceleration due to gravity were different?
2. What would the maximum range be on the Moon, where gravity is 1/6th of Earth's gravity?
3. How long does the ball stay in the air when thrown vertically?
4. How much time does the ball take to reach its maximum height when thrown vertically?
5. What is the speed of the ball at its highest point when thrown vertically?

**Tip:** The angle of \( 45^\circ \) is ideal for achieving the maximum range in projectile motion, assuming no air resistance.

Learn how to calculate the maximum height a boy can throw a ball vertically upward if he can throw it 52 meters horizontally. This problem involves projectile motion, kinematics, and the relationship between range and height. Suitable for students in Grades 9-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation