A stone is launched straight up by a slingshot with an initial speed of 19.6 m/s. The stone is located 1.50 m above the ground when launched. How high, above the ground, does the stone rise?

We can solve this problem using the kinematic equations for motion under constant acceleration. The stone is subject to gravity, which acts as a deceleration with a value of \( g = 9.8 \, \text{m/s}^2 \).

We need to find the maximum height the stone reaches above the ground. At maximum height, the final velocity of the stone will be zero because it momentarily stops before falling back down. 

We can use the following kinematic equation to solve for the displacement (\( \Delta y \)) from the point of launch to the maximum height:

\[
v^2 = u^2 + 2a\Delta y
\]

Where:
- \( v = 0 \, \text{m/s} \) is the final velocity at the maximum height,
- \( u = 19.6 \, \text{m/s} \) is the initial velocity,
- \( a = -9.8 \, \text{m/s}^2 \) is the acceleration due to gravity (negative because it opposes the motion),
- \( \Delta y \) is the vertical displacement from the launch point to the maximum height.

### Step 1: Solve for \( \Delta y \)

Rearrange the equation to solve for \( \Delta y \):

\[
0 = (19.6)^2 + 2(-9.8)\Delta y
\]
\[
0 = 384.16 - 19.6 \Delta y
\]
\[
19.6 \Delta y = 384.16
\]
\[
\Delta y = \frac{384.16}{19.6} = 19.6 \, \text{m}
\]

This is the distance the stone travels upwards from the launch point to its maximum height.

### Step 2: Find the total height above the ground

The stone is launched from a height of 1.50 m above the ground, so the total height above the ground is:

\[
\text{Total height} = 19.6 \, \text{m} + 1.50 \, \text{m} = 21.10 \, \text{m}
\]

### Final Answer:
The stone rises to a height of **21.10 meters** above the ground.

Would you like further details or have any questions?  
Here are some related questions:

1. How long does it take for the stone to reach its maximum height?
2. What is the total time the stone stays in the air?
3. What is the stone’s velocity when it hits the ground?
4. How would the maximum height change if the initial speed was increased to 25 m/s?
5. How does the angle of launch affect the maximum height for projectile motion?

**Tip:** In problems involving motion, make sure to pay attention to the direction of acceleration and velocity—gravity always acts downward!

Solve how high a stone rises when launched upwards with a speed of 19.6 m/s from 1.50 m above the ground. This problem covers projectile motion and the kinematic equation for uniformly accelerated motion, making it ideal for students in Grades 9-11.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation