Billy Joe B. throws a rock at a tree because… reasons. He throws it 37.43m, and the rock takes 2.5s to reach the tree where it hits the ground right in front of it. What is the magnitude of the rock's final velocity if this is an angled projectile?



To solve for the magnitude of the rock's final velocity in this projectile motion scenario, we can break the motion into horizontal and vertical components and use the principles of kinematics.

### Step 1: Determine horizontal and vertical components of the motion.

#### Horizontal Motion:
The horizontal distance (\( x \)) is given as 37.43 m, and the time of flight (\( t \)) is 2.5 s. The horizontal velocity (\( v_x \)) remains constant throughout the flight (assuming no air resistance).

\[
v_x = \frac{x}{t} = \frac{37.43}{2.5} = 14.972 \, \text{m/s}
\]

#### Vertical Motion:
The vertical component of the motion involves constant acceleration due to gravity (\( g = 9.8 \, \text{m/s}^2 \)). However, we need to find the vertical velocity just before the rock hits the ground.

The total time of flight is 2.5 seconds, and we assume the rock lands at the same vertical height from which it was thrown (since it hits the ground). The initial vertical velocity (\( v_{y0} \)) can be found using symmetry, but for now, we focus on finding the final vertical velocity.

Using the kinematic equation for vertical velocity:

\[
v_y = v_{y0} - g t
\]

If the initial vertical velocity is zero, then:

\[
v_y = - g t = -9.8 \times 2.5 = -24.5 \, \text{m/s}
\]

(Note that the negative sign indicates the downward direction.)

### Step 2: Calculate the magnitude of the final velocity.

The final velocity is a vector composed of the horizontal (\( v_x \)) and vertical (\( v_y \)) components. To find the magnitude of the final velocity (\( v_f \)):

\[
v_f = \sqrt{v_x^2 + v_y^2}
\]

Substituting the values:

\[
v_f = \sqrt{(14.972)^2 + (-24.5)^2} = \sqrt{224.16 + 600.25} = \sqrt{824.41} \approx 28.71 \, \text{m/s}
\]

### Final Answer:
The magnitude of the rock's final velocity is approximately **28.71 m/s**.

Would you like more details or clarifications on any of these steps?

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Here are five related questions:
1. How would the final velocity change if the initial vertical velocity was non-zero?
2. What would happen if the rock was thrown from a higher point above the ground?
3. How can we calculate the initial velocity angle in this scenario?
4. How does air resistance affect projectile motion?
5. How can we determine the total energy of the rock at the moment it hits the tree?

**Tip:** In projectile motion, the horizontal component of velocity remains constant, while the vertical component changes due to gravity.

Billy Joe B. throws a rock at a tree because… reasons. He throws it 37.43m, and the rock takes 2.5s to reach the tree where it hits the ground right in front of it. What is the magnitude of the rock's final velocity if this is an angled projectile?

Discover how to calculate the final velocity of a rock thrown at 37.43m over 2.5s in this projectile motion problem. Learn about horizontal and vertical components, using key kinematic formulas. Suitable for high school students in Grades 9-12.

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