A 2.0 kg object initially at rest at the origin is subjected to the time-varying force shown in FIGURE P6.38. What is the object's velocity at t = 4 s?

Verified step by step guidance

1. Step 1: Understand the problem. The object starts at rest, meaning its initial velocity is zero. The force applied to the object varies with time, as shown in the graph. To find the velocity at t = 4 s, we need to calculate the impulse delivered to the object up to t = 4 s and use the impulse-momentum theorem.
2. Step 2: Recall the impulse-momentum theorem, which states that the impulse (J) is equal to the change in momentum (Δp). The impulse is given by the integral of the force over time: $J={\int }_t^{F}dt$. Since the object starts at rest, its initial momentum is zero, and the final momentum is equal to the impulse.
3. Step 3: Break the graph into segments to calculate the impulse. From t = 0 to t = 2 s, the force increases linearly from 0 N to 4 N. The impulse for this segment can be calculated as the area of the triangle under the curve: $J=\frac{1}{2}\times \mathrm{base}\times \mathrm{height}$.
4. Step 4: From t = 2 s to t = 4 s, the force remains constant at 4 N. The impulse for this segment is the area of the rectangle under the curve: $J=\mathrm{force}\times \mathrm{time}$. Add this impulse to the previous segment's impulse.
5. Step 5: Use the total impulse to find the velocity. The momentum is given by $p=m\times v$, where m is the mass of the object. Rearrange to solve for velocity: $v=\frac{J}{m}$. Substitute the total impulse and the mass (2.0 kg) to find the velocity at t = 4 s.

Key Concepts

Newton's Second Law of Motion

Newton's Second Law states that the acceleration of an object is directly proportional to the net force acting on it and inversely proportional to its mass. This relationship is expressed mathematically as F = ma, where F is the net force, m is the mass, and a is the acceleration. Understanding this law is crucial for analyzing how the force applied to the object affects its motion.

Impulse and Momentum

Impulse is defined as the change in momentum of an object when a force is applied over a period of time. It is calculated as the product of the average force and the time duration over which it acts. The relationship between impulse and momentum is given by the equation Impulse = Δp = F_avg * Δt, where Δp is the change in momentum. This concept is essential for determining the object's velocity after the force has been applied.

Area Under the Force-Time Graph

The area under a force-time graph represents the impulse delivered to an object. In this case, the graph shows a time-varying force applied to the object, and calculating the area under the curve from t = 0 to t = 4 seconds will provide the total impulse. This impulse can then be used to find the change in momentum and, consequently, the final velocity of the object.