You throw a nerf basketball of mass 0.002 kg into the air. At t = 0, the ball leaves your hand. At t = 0.10 s, the net force on the ball is

(-0.0184, -0.0224, 0) N, its velocity is (1.87, 0.84, 0) m/s, and its position is (0.325, 0.180, 0) m. (Besides the gravitational force by Earth, air also exerts a force on the ball.)

if rnew = (0.35872, 0.19232,0). then what will the new position be if t=0.12 s, the net force on the ball is now (-0.0138, -0.0197, 0) N. What will be its position at t = 0.14 s?

To determine the new position of the nerf basketball at \( t = 0.12 \) s and \( t = 0.14 \) s, we'll use the principles of kinematics and Newton's second law. Here's the plan:

1. **Calculate the acceleration** at \( t = 0.10 \) s and \( t = 0.12 \) s using the net forces provided.
2. **Find the velocity** of the ball at \( t = 0.12 \) s and \( t = 0.14 \) s.
3. **Determine the position** of the ball at \( t = 0.14 \) s using the velocity at \( t = 0.12 \) s and the acceleration.

### Step 1: Calculate Acceleration
Using Newton's second law:

\[
\vec{a} = \frac{\vec{F}_{\text{net}}}{m}
\]

For \( t = 0.10 \) s:
- \(\vec{F}_{\text{net}} = (-0.0184, -0.0224, 0)\) N
- \(m = 0.002\) kg

For \( t = 0.12 \) s:
- \(\vec{F}_{\text{net}} = (-0.0138, -0.0197, 0)\) N

### Step 2: Update Velocity
The velocity at \( t = 0.12 \) s can be calculated using:

\[
\vec{v}_{\text{new}} = \vec{v}_{\text{old}} + \vec{a} \Delta t
\]

Similarly, for \( t = 0.14 \) s, use the velocity from \( t = 0.12 \) s.

### Step 3: Update Position
The new position can be found using:

\[
\vec{r}_{\text{new}} = \vec{r}_{\text{old}} + \vec{v} \Delta t
\]

We'll compute these values step by step.The position of the nerf basketball at \( t = 0.14 \) s will be approximately \((0.390, 0.201, 0.0)\) meters.

Would you like further details on the calculations, or do you have any additional questions?

Here are five related questions you might consider:
1. How would the position change if the air resistance was different?
2. Can you explain how the force of gravity is integrated into these calculations?
3. What would happen if the ball was thrown vertically instead?
4. How does air resistance affect the motion of the ball over time?
5. What is the significance of using small time intervals in such calculations?

**Tip:** When dealing with forces and motion, breaking the problem into small time steps can significantly improve the accuracy of the results.

Learn how to calculate the position of a nerf basketball thrown into the air using forces and kinematics. Solve for positions at specific times based on given forces and initial conditions. Suitable for advanced high school students studying Newton's laws of motion and kinematics.

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