Answer:
we can use the work-energy principle. The work done on the box by the applied force and the work done by friction will change the kinetic energy of the box, which can be used to find the final velocity.
Explanation:
To find the final velocity of the box, we can use the work-energy principle. The work done on the box by the applied force and the work done by friction will change the kinetic energy of the box, which can be used to find the final velocity.
First, let's resolve the applied force into horizontal and vertical components. The horizontal component \( F_{\text{horizontal}} \) is given by \( F \cos(\theta) \), and the vertical component \( F_{\text{vertical}} \) is given by \( F \sin(\theta) \), where \( F \) is the applied force and \( \theta \) is the angle of application.
Given:
- Mass of the box (\( m \)) = 12.3 kg
- Applied force (\( F \)) = 34.6 N
- Angle of application (\( \theta \)) = 42.9°
- Frictional force (\( f \)) = 9.63 N
- Distance moved (\( d \)) = 7.2 m
The horizontal component of the applied force is:
\[ F_{\text{horizontal}} = F \cos(\theta) = 34.6 \times \cos(42.9°) \]
The work done by the applied force (\( W_{\text{applied}} \)) is:
\[ W_{\text{applied}} = F_{\text{horizontal}} \times d \]
The work done by friction (\( W_{\text{friction}} \)) is:
\[ W_{\text{friction}} = -f \times d \]
The net work done (\( W_{\text{net}} \)) is the sum of the work done by the applied force and the work done by friction:
\[ W_{\text{net}} = W_{\text{applied}} + W_{\text{friction}} \]
This net work done will be equal to the change in kinetic energy (\( \Delta KE \)) of the box:
\[ W_{\text{net}} = \Delta KE = \frac{1}{2} m v^2 - \frac{1}{2} m u^2 \]
Since the box starts from rest, the initial velocity (\( u \)) is 0, so:
\[ W_{\text{net}} = \frac{1}{2} m v^2 \]
Solving for the final velocity (\( v \)), we get:
\[ v = \sqrt{\frac{2 W_{\text{net}}}{m}} \]
Now, plug in the values and calculate \( F_{\text{horizontal}} \), \( W_{\text{applied}} \), \( W_{\text{friction}} \), and \( W_{\text{net}} \) to find the final velocity \( v \). Remember to convert the angle to radians if you are using a calculator that requires it. The final velocity will be in meters per second (m/s).
