Understanding the Problem
In this scenario, we are analyzing a $7\text{ kg}$ wagon moving across a frictionless horizontal surface. We are given the following information:
- Mass ($m$): $7\text{ kg}$
- Initial velocity ($u$): $4\text{ m/s}$
- Applied Force ($F$): $10\text{ N}$
- Distance covered ($d$): $3\text{ m}$
We need to find:
- The acceleration ($a$) produced by the force.
- The final speed ($v$) of the wagon.
Step 1: Calculating Acceleration
Using Newton's Second Law, which states that the net force applied to an object is equal to the product of its mass and its acceleration ($F = ma$):
$$a = \frac{F}{m}$$ $$a = \frac{10\text{ N}}{7\text{ kg}} \approx 1.43\text{ m/s}^2$$
Step 2: Calculating Final Speed
We can find the final velocity using the kinematic equation of motion, specifically $v^2 = u^2 + 2ad$:
$$v^2 = (4\text{ m/s})^2 + 2 \times (1.43\text{ m/s}^2) \times (3\text{ m})$$ $$v^2 = 16 + 8.58$$ $$v^2 = 24.58$$ $$v = \sqrt{24.58} \approx 4.96\text{ m/s}$$
Alternative Approach: The Work-Energy Theorem
The Work-Energy Theorem states that the work done by a net force equals the change in kinetic energy:
$$W = \Delta KE = \frac{1}{2}mv^2 - \frac{1}{2}mu^2$$
Given $W = F \times d = 10\text{ N} \times 3\text{ m} = 30\text{ J}$:
$$30 = \frac{1}{2} \times 7 \times (v^2 - 16)$$ $$60 = 7(v^2 - 16)$$ $$8.57 = v^2 - 16$$ $$v^2 = 24.57$$ $$v \approx 4.96\text{ m/s}$$
Both methods yield the same result, confirming our calculation is correct.