The Problem

A stone is thrown vertically at a speed of 30 m/s, making an angle of 45 degrees with the horizontal. We need to determine the maximum height reached by the stone.

Solution

To solve this problem, we can break it down into two components: the vertical component and the horizontal component of the stone's velocity.

Vertical Component of the Velocity

The vertical component of the velocity can be calculated using the formula:

v_vertical = v_initial * sin(theta)

where v_initial is the initial velocity of the stone and theta is the angle with the horizontal.

In this case, the initial velocity is 30 m/s and the angle is 45 degrees.

v_vertical = 30 m/s * sin(45 degrees)
v_vertical = 30 m/s * 0.7071
v_vertical ≈ 21.213 m/s

Time Taken to Reach Maximum Height

The time taken to reach maximum height can be calculated using the formula:

t = v_vertical / g

where g is the acceleration due to gravity (approximately 9.8 m/s^2).

t = 21.213 m/s / 9.8 m/s^2
t ≈ 2.165 seconds

Maximum Height Reached

The maximum height reached by the stone can be calculated using the formula:

h = v_vertical * t - (1/2) * g * t^2

h = 21.213 m/s * 2.165 s - (1/2) * 9.8 m/s^2 * (2.165 s)^2
h ≈ 22.959 meters

Therefore, the maximum height reached by the stone is approximately 22.959 meters.

Summary

- The stone is thrown vertically at a speed of 30 m/s, making an angle of 45 degrees with the horizontal.
- The vertical component of the velocity is calculated using the formula v_vertical = v_initial * sin(theta).
- The time taken to reach maximum height is calculated using the formula t = v_vertical / g.
- The maximum height reached by the stone is calculated using the formula h = v_vertical * t - (1/2) * g * t^2.
- The maximum height reached by the stone is approximately 22.959 meters.