https://file.aiquickdraw.com/mathbot/file/1730170066311ocn70vuz.jpg

To solve this problem, we’ll need to calculate the change in velocity of the object due to the force applied over the given time.

### Given Data
- Mass of the object, \( m = 2.1 \, \text{kg} \)
- Initial velocity, \( v_{\text{initial}} = 1.0 \, \text{m/s} \) (to the right)
- The force \( F_x(t) \) applied to the object varies over a time interval of \( t = 0.50 \, \text{s} \).

### Step 1: Calculate Impulse
Impulse (\( J \)) is the area under the force-time graph, and it represents the change in momentum. The force-time graph shows a triangular area, with:
- Base \( = 0.50 \, \text{s} \)
- Height \( = -12 \, \text{N} \)

The area of a triangle is given by:
\[
\text{Area} = \frac{1}{2} \times \text{base} \times \text{height}
\]
Substituting values:
\[
J = \frac{1}{2} \times 0.50 \, \text{s} \times (-12 \, \text{N}) = -3.0 \, \text{Ns}
\]

### Step 2: Calculate Change in Velocity
Impulse is also given by:
\[
J = \Delta p = m \Delta v
\]
where \( \Delta v \) is the change in velocity. Solving for \( \Delta v \):
\[
\Delta v = \frac{J}{m} = \frac{-3.0 \, \text{Ns}}{2.1 \, \text{kg}} \approx -1.43 \, \text{m/s}
\]

### Step 3: Find Final Velocity
Since the initial velocity is \( 1.0 \, \text{m/s} \) to the right, and the impulse decreases the velocity, the final velocity \( v_{\text{final}} \) is:
\[
v_{\text{final}} = v_{\text{initial}} + \Delta v = 1.0 \, \text{m/s} - 1.43 \, \text{m/s} = -0.43 \, \text{m/s}
\]

### Conclusion
The final velocity of the object is approximately \( -0.43 \, \text{m/s} \), indicating it moves to the left after the force ends.

Would you like more details on any of these steps or have any questions?

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### Related Questions
1. How would the result change if the object's mass were different?
2. What would happen if the initial velocity were directed to the left?
3. How is impulse affected if the force-time graph is a rectangle instead of a triangle?
4. How do we determine the direction of motion after the impulse?
5. How does the impulse-momentum theorem apply in collision problems?

### Tip
When analyzing impulse, always consider both the magnitude and direction of the force applied over time.

What are the object's speed and direction after the force ends?

Determine the final speed and direction of a 2.1 kg object after experiencing a force impulse over 0.5 s. This problem uses the Impulse-Momentum Theorem, analyzing the area under a force-time graph to find the object's final velocity. Ideal for Grades 11-12 students.

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