To find the force responsible for the acceleration of the shopping cart, we can use Newton's second law of motion, which states:
\[ F = m \cdot a \]
where:
- \( F \) is the force,
- \( m \) is the mass, and
- \( a \) is the acceleration.
First, we need to calculate the acceleration (\( a \)). We can use the formula for acceleration:
\[ a = \frac{\Delta v}{\Delta t} \]
where:
- \( \Delta v \) is the change in velocity,
- \( \Delta t \) is the change in time.
Given:
- Initial velocity (\( v_i \)) = 0 m/s (starting from rest),
- Final velocity (\( v_f \)) = 4.0 m/s,
- Time (\( \Delta t \)) = 0.5 seconds.
Now, we can calculate the acceleration:
\[ \Delta v = v_f - v_i = 4.0 , \text{m/s} - 0 , \text{m/s} = 4.0 , \text{m/s} \]
Now, inserting the values into the acceleration formula:
\[ a = \frac{4.0 , \text{m/s}}{0.5 , \text{s}} = 8.0 , \text{m/s}^2 \]
Now that we have the acceleration, we can find the force using the mass of the shopping cart:
Given:
- Mass (\( m \)) = 8.0 kg.
Using the formula:
\[ F = m \cdot a = 8.0 , \text{kg} \cdot 8.0 , \text{m/s}^2 = 64.0 , \text{N} \]
Thus, the magnitude of the force responsible for the acceleration of the shopping cart is 64.0 N.