Problem 27 - University Physics with Modern Physics

Question

A little red wagon with mass 7.00 \(\mathrm{kg}\) moves in a straight line on a frictionless horizontal surface. It has an initial speed of 4.00 \(\mathrm{m} / \mathrm{s}\) and then is pushed 3.0 \(\mathrm{m}\) in the direction of the initial velocity by a force with a magnitude of 10.0 \(\mathrm{N}\) . (a) Use the work-energy theorem to calculare the wagon's final speed. (b) Cal- culate the acceleration produced by the force. Use this acceleration in the kinematic relationships of Chapter 2 to calculate the wagon's final speed. Compare this result to that calculated in part (a).

Step-by-Step Solution

Answer

The wagon's final speed is 4.95 m/s using both methods.

Step 1: Identify Given Values

First, gather all the relevant data from the problem. The mass of the wagon is 7.00 \(\mathrm{kg}\), the initial speed \(v_i\) is 4.00 \(\mathrm{m/s}\), the displacement \(s\) is 3.0 \(\mathrm{m}\), and the force \(F\) applied is 10.0 \(\mathrm{N}\).

Step 2: Apply the Work-Energy Theorem

The work-energy theorem states that the work done on an object is equal to the change in kinetic energy. Calculate the work done by the force: \[ W = F \times s = 10.0 \mathrm{\, N} \times 3.0 \mathrm{\, m} = 30.0 \mathrm{\, J} \]

Step 3: Calculate the Change in Kinetic Energy

The change in kinetic energy \( \Delta KE \) is equal to the work done on the wagon. Thus, \( \Delta KE = W = 30.0 \mathrm{\, J} \). The initial kinetic energy \( KE_i = \frac{1}{2}mv_i^2 = 0.5 \times 7.00 \times 4.00^2 \).

Step 4: Find Initial and Final Kinetic Energies

Calculate the initial kinetic energy: \[ KE_i = 0.5 \times 7.00 \times 4.00^2 = 56.0 \mathrm{\, J} \] Use \( \Delta KE = KE_f - KE_i = 30.0 \mathrm{\, J} \) to find the final kinetic energy: \( KE_f = KE_i + \Delta KE = 56.0 + 30.0 = 86.0 \mathrm{\, J} \).

Step 5: Calculate the Final Velocity (Part a)

Use the final kinetic energy to find the final speed \(v_f\). \[ KE_f = \frac{1}{2}mv_f^2 = 86.0 \mathrm{\, J} \rightarrow \frac{1}{2}\times7.00\times v_f^2 = 86.0 \] Solve for \(v_f\): \[ v_f = \sqrt{\frac{2\times86.0}{7.00}} = 4.95 \mathrm{\, m/s} \].

Step 6: Calculate Acceleration (Part b)

Using Newton's second law \( F = m\times a \) to find acceleration \(a\): \[ a = \frac{F}{m} = \frac{10.0}{7.0} = 1.43 \mathrm{\, m/s^2} \].

Step 7: Use Kinematic Equations to Find Final Velocity

With acceleration known, use the kinematic equation \(v_f^2 = v_i^2 + 2as\) to find \(v_f\). \[ v_f^2 = 4.00^2 + 2\times1.43\times3.0 \] \[ v_f = \sqrt{16 + 8.58} = 4.95 \mathrm{\, m/s} \].

Step 8: Compare Results

The final velocities from both methods are 4.95 \(\mathrm{m/s}\). This shows consistency between the work-energy method and the kinematic approach.

Key Concepts

Work-Energy Theorem

The Work-Energy Theorem is a powerful principle in physics that connects the work done on an object to its kinetic energy. This theorem states that the work done by all external forces acting on an object is equal to the change in its kinetic energy. Mathematically, it can be expressed as:

• \[ W = \Delta KE \]

In the context of our red wagon problem, the work done by the 10.0 N force over a displacement of 3.0 m is calculated as 30.0 J. Since the surface is frictionless, this work directly translates into changing the kinetic energy of the wagon.

The initial kinetic energy (\( KE_i \)) can be determined using the formula \( KE = \frac{1}{2}mv^2 \). For the initial speed (4.0 m/s) and mass (7.0 kg), the initial kinetic energy is 56.0 J. The work done (30.0 J) is added to this initial kinetic energy to find the final kinetic energy (86.0 J). From here, we can re-arrange the kinetic energy formula to solve for the final speed, confirming it as 4.95 m/s.

Kinematic Equations

Kinematic equations are equations that describe the motion of objects using variables such as velocity, acceleration, displacement, and time. One of the most commonly used kinematic equations is:

• \[ v_f^2 = v_i^2 + 2as \]

In the problem at hand, once the acceleration is known, this equation becomes key in confirming the final speed calculated via the work-energy theorem.

The initial speed \( v_i \) is 4.0 m/s, and the acceleration \( a \) is determined to be 1.43 m/s². Plugging in the values, and the displacement \( s = 3.0 \) m, the kinematic equation also gives a final velocity \( v_f \) of 4.95 m/s.

This confirms the consistency of the results from the two different approaches in analyzing the same physical situation.

Newton's Second Law

Newton's Second Law of Motion is fundamental to solving many physics problems. It tells us that the force acting on an object is equal to the mass of the object multiplied by its acceleration.

• \[ F = ma \]

For the red wagon, this law is used to calculate the acceleration produced by the applied force. With a force of 10.0 N and a mass of 7.0 kg, the acceleration can be found using the rearranged form of Newton's Second Law:

• \[ a = \frac{F}{m} = \frac{10.0}{7.0} \approx 1.43 \mathrm{\, m/s^2} \]

This calculated acceleration is critical in using the kinematic equations to determine the wagon's final speed.

By understanding Newton's Second Law, you can appreciate how forces influence the motion of objects, contributing to the cornerstone of dynamic analysis in physics.