Part 1: Force Applied to the Right

The force $F = 25.0 \, \text{N}$ is applied for $\Delta t = 0.050 \, \text{s}$, and the puck's mass is $m = 0.160 \, \text{kg}$.

Step 1: Calculate the acceleration

Using Newton's second law: $a = \frac{F}{m}$ $a = \frac{25.0 \, \text{N}}{0.160 \, \text{kg}} = 156.25 \, \text{m/s}^2$

Step 2: Calculate the change in velocity

The change in velocity due to this acceleration over the given time is: $\Delta v = a \cdot \Delta t$ $\Delta v = 156.25 \, \text{m/s}^2 \cdot 0.050 \, \text{s} = 7.8125 \, \text{m/s}$

Step 3: Calculate the final velocity

The initial velocity is $v_0 = 3.00 \, \text{m/s}$, and the force is applied in the same direction as the initial motion: $v_f = v_0 + \Delta v$ $v_f = 3.00 \, \text{m/s} + 7.8125 \, \text{m/s} = 10.8125 \, \text{m/s}$

Thus, the final velocity is: $\boxed{10.8 \, \text{m/s} \text{ (to the right)}}$

---

Part 2: Force Applied to the Left

The force $F = 12.0 \, \text{N}$ is applied in the opposite direction to the initial motion, over the same time interval.

Step 1: Calculate the acceleration

Using Newton's second law: $a = \frac{F}{m}$ $a = \frac{12.0 \, \text{N}}{0.160 \, \text{kg}} = 75.0 \, \text{m/s}^2$

Step 2: Calculate the change in velocity

The change in velocity is: $\Delta v = a \cdot \Delta t$ $\Delta v = 75.0 \, \text{m/s}^2 \cdot 0.050 \, \text{s} = 3.75 \, \text{m/s}$

Step 3: Calculate the final velocity

Since the force is applied to the left, the velocity change is negative relative to the initial motion: $v_f = v_0 - \Delta v$ $v_f = 3.00 \, \text{m/s} - 3.75 \, \text{m/s} = -0.75 \, \text{m/s}$

A negative velocity indicates motion to the left. Thus, the final velocity is: $\boxed{0.75 \, \text{m/s} \text{ (to the left)}}$

---

Summary:

1. Final velocity with a 25.0 N force to the right: $\boxed{10.8 \, \text{m/s} \, \text{(to the right)}}$
2. Final velocity with a 12.0 N force to the left: $\boxed{0.75 \, \text{m/s} \, \text{(to the left)}}$

---

Would you like more detailed steps or explanations?

Related Questions:

1. How would the final velocity change if the force to the right was doubled?
2. How long would it take for the puck to stop if only the leftward force was applied?
3. What is the impulse imparted to the puck in each case?
4. If the surface was not frictionless, how would the results change?
5. How much work is done by the applied force in each scenario?

Tip:

To calculate impulse, remember the formula: $\text{Impulse} = F \cdot \Delta t = \Delta p$ where $\Delta p = m \cdot \Delta v$.