An autographed baseball rolls off of a 0.61 m
high desk and strikes the floor 0.10 m away
from the desk.
How fast was it rolling on the desk before
it fell off? The acceleration of gravity is
9.81 m/s
2
.
Answer in units of m/s.

To solve this problem, we can use the principles of projectile motion. Since the ball rolls horizontally off the desk, its initial vertical velocity is zero, and it only has an initial horizontal velocity that we are trying to find.

### Step 1: Calculate the time it takes to fall to the ground.

1. **Vertical distance (height)** from the desk to the floor: \( h = 0.61 \, \text{m} \)
2. **Acceleration due to gravity**: \( g = 9.81 \, \text{m/s}^2 \)
3. Since the initial vertical velocity \( v_{0y} = 0 \), we use the following equation for vertical motion:
   \[
   h = \frac{1}{2} g t^2
   \]
4. Solving for \( t \):
   \[
   t = \sqrt{\frac{2h}{g}}
   \]

Substituting the values:
\[
t = \sqrt{\frac{2 \times 0.61}{9.81}} = \sqrt{\frac{1.22}{9.81}} \approx \sqrt{0.1244} \approx 0.353 \, \text{s}
\]

So, the time of flight is approximately \( t = 0.353 \, \text{s} \).

### Step 2: Calculate the horizontal velocity.

Now, we know that the ball traveled horizontally \( x = 0.10 \, \text{m} \) during this time. We can use the formula for horizontal motion:
\[
x = v_x \cdot t
\]
Solving for \( v_x \):
\[
v_x = \frac{x}{t}
\]

Substituting the values:
\[
v_x = \frac{0.10}{0.353} \approx 0.283 \, \text{m/s}
\]

### Answer
The speed of the ball as it was rolling on the desk before it fell off was approximately:
\[
\boxed{0.283 \, \text{m/s}}
\]

Would you like further clarification or have any questions?

---

Here are some related questions that could deepen your understanding:
1. How would the answer change if the desk were higher or lower?
2. What would happen if there were an initial vertical velocity component?
3. How can we apply these principles to other objects rolling off surfaces?
4. What assumptions are made in projectile motion problems like this?
5. How does air resistance affect the motion of the baseball in real-life scenarios?

**Tip:** When dealing with projectile motion, always separate the horizontal and vertical components of motion, as they are independent of each other.

An autographed baseball rolls off of a 0.61 m high desk and strikes the floor 0.10 m away from the desk. How fast was it rolling on the desk before it fell off? The acceleration of gravity is 9.81 m/s^2. Answer in units of m/s.

Learn how to calculate the horizontal speed of a baseball rolling off a desk using principles of projectile motion. The problem involves calculating the time of flight and horizontal velocity. Given a height of 0.61 m and a horizontal distance of 0.10 m, the speed is found to be approximately 0.283 m/s.

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