A 49.1 kg wagon is towed up a hill inclined
at 17.8◦ with respect to the horizontal. The
tow rope is parallel to the incline and has a
tension of 158 N in it. Assume that the wagon
starts from rest at the bottom of the hill, and
neglect friction.
The acceleration of gravity is 9.8 m/s^2
How fast is the wagon going after moving
54.5 m up the hill? Answer in units of m/s.

We can solve this problem using the work-energy principle. Here's the step-by-step process:

### Step 1: Given information
- Mass of the wagon: \( m = 49.1 \, \text{kg} \)
- Incline angle: \( \theta = 17.8^\circ \)
- Tension in the rope: \( T = 158 \, \text{N} \)
- Distance traveled up the incline: \( d = 54.5 \, \text{m} \)
- Gravitational acceleration: \( g = 9.8 \, \text{m/s}^2 \)
- Initial velocity: \( v_0 = 0 \, \text{m/s} \) (since the wagon starts from rest)

### Step 2: Net force along the incline
The forces acting along the incline include:
- The tension force pulling the wagon up, which is \( T = 158 \, \text{N} \).
- The gravitational force component pulling the wagon down, which is \( m \cdot g \cdot \sin(\theta) \).

The net force \( F_{\text{net}} \) acting on the wagon is:

\[
F_{\text{net}} = T - m \cdot g \cdot \sin(\theta)
\]

### Step 3: Net acceleration
Using Newton's second law, the net force is related to acceleration \( a \) by:

\[
F_{\text{net}} = m \cdot a
\]

Thus, the acceleration is:

\[
a = \frac{F_{\text{net}}}{m} = \frac{T - m \cdot g \cdot \sin(\theta)}{m}
\]

### Step 4: Final velocity using kinematic equation
We can use the following kinematic equation to find the final velocity \( v \):

\[
v^2 = v_0^2 + 2ad
\]

Since the wagon starts from rest (\( v_0 = 0 \)):

\[
v = \sqrt{2ad}
\]

Now let's calculate these values.The wagon's speed after moving 54.5 meters up the hill is approximately \( 4.92 \, \text{m/s} \).

Would you like more details or have any questions? Here are 5 related questions to further your understanding:

1. How would the result change if the incline angle was steeper?
2. What would happen if friction was not neglected?
3. How does the tension in the rope compare to the weight of the wagon?
4. How could we determine the time it takes for the wagon to reach the final velocity?
5. What is the total work done by the rope on the wagon?

**Tip:** Remember, when analyzing inclined planes, always break down forces into parallel and perpendicular components relative to the slope.

A 49.1 kg wagon is towed up a hill inclined at 17.8° with respect to the horizontal. The tow rope is parallel to the incline and has a tension of 158 N in it. Assume that the wagon starts from rest at the bottom of the hill, and neglect friction. The acceleration of gravity is 9.8 m/s². How fast is the wagon going after moving 54.5 m up the hill? Answer in units of m/s.

Learn how to calculate the final velocity of a 49.1 kg wagon towed up a 17.8° inclined hill using Newton's Second Law and the Work-Energy Principle. With given tension and gravitational force, we break down the forces and solve for the velocity after moving 54.5 meters. Suitable for Grades 11-12.

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