Let's break down the problem step by step and solve it systematically.
### Step 1: Mass of the Cart
We are given the following information:
- A constant net force [tex]\( F_{\text{net}} = 200 \)[/tex] N is exerted.
- The cart accelerates from rest, so the initial velocity [tex]\( u = 0 \)[/tex] m/s.
- The final velocity [tex]\( v = 4 \)[/tex] m/s.
- We need to find the mass of the cart [tex]\( m_{\text{cart}} \)[/tex].
#### Step 1.1: Calculate the acceleration of the cart
Use the formula for acceleration:
[tex]\[ a = \frac{v - u}{t} \][/tex]
Assuming the time [tex]\( t \)[/tex] taken for this change in velocity is 1 second (for simplicity), the acceleration [tex]\( a \)[/tex] can be calculated as:
[tex]\[ a = \frac{4 \text{ m/s} - 0 \text{ m/s}}{1 \text{ s}} = 4 \text{ m/s}^2 \][/tex]
#### Step 1.2: Calculate the mass of the cart
Use Newton's second law, which states that force is the product of mass and acceleration:
[tex]\[ F = ma \][/tex]
Rearranging for mass:
[tex]\[ m = \frac{F}{a} \][/tex]
Substitute the given values:
[tex]\[ m_{\text{cart}} = \frac{200 \text{ N}}{4 \text{ m/s}^2} = 50 \text{ kg} \][/tex]
So, the mass of the cart is 50 kg.
### Step 2: Acceleration of the Ball
We are given the following information:
- The mass of the ball [tex]\( m_{\text{ball}} = 0.40 \)[/tex] kg.
- A force [tex]\( F_{\text{ball}} = 200 \)[/tex] N is exerted on the ball.
#### Step 2.1: Calculate the acceleration of the ball
Again, use Newton's second law:
[tex]\[ F_{\text{ball}} = m_{\text{ball}} \cdot a_{\text{ball}} \][/tex]
Rearrange to find the acceleration:
[tex]\[ a_{\text{ball}} = \frac{F_{\text{ball}}}{m_{\text{ball}}} \][/tex]
Substitute the given values:
[tex]\[ a_{\text{ball}} = \frac{200 \text{ N}}{0.40 \text{ kg}} = 500 \text{ m/s}^2 \][/tex]
So, the acceleration of the ball is 500 m/s².
### Conclusion
1. The mass of the cart is 50 kg.
2. The acceleration of the ball is 500 m/s².
