Problem 22 - College Physics

Question

The engine of a 1.0 -kg toy plane exerts a 15-N forward force. If the air exerts an 8.0 -N resistive force on the plane, what is the magnitude of the acceleration of the plane?

Step-by-Step Solution

Answer

The acceleration of the plane is 7 m/s².

Step 1: Understanding the problem

To solve the problem, we need to find the net force acting on the toy plane and then use this to determine the plane's acceleration. The toy plane has a forward force of 15 N and a resistive force of 8 N opposing this forward force.

Step 2: Calculate the net force

The net force on an object is the vector sum of all forces acting on it. Here, the forward force is 15 N and the resistive force is 8 N. The net force can be calculated as follows:\[ F_{net} = F_{forward} - F_{resistive} = 15 ext{ N} - 8 ext{ N} = 7 ext{ N} \]

Step 3: Apply Newton's Second Law

According to Newton's Second Law, the acceleration of an object can be determined using the formula:\[ F = m imes a \]where \( F \) is the net force, \( m \) is the mass, and \( a \) is the acceleration. Rearrange to solve for \( a \):\[ a = \frac{F}{m} \]

Step 4: Calculate the acceleration

Substitute the known values into the equation:\[ a = \frac{7 ext{ N}}{1.0 ext{ kg}} = 7 ext{ m/s}^2 \]

Step 5: Conclusion

The magnitude of the acceleration of the toy plane is 7 m/s².

Key Concepts

Net Force Calculation

In physics, understanding net force is crucial for determining how objects move. When two or more forces act upon an object, these forces can be combined to find the net force. This is simply the total force taking direction into account.

For our toy plane, we have two forces at play: a forward force of 15 N propelling the plane and a resistive force of 8 N working against it. To calculate the net force, we subtract the resistive force from the forward force:

• The forward force is positive: 15 N.
• The resistive force is negative as it opposes the forward force: -8 N.
• Net force: 15 N - 8 N = 7 N.

This indicates a net force of 7 N in the direction of the forward force. By understanding and calculating the net force, we can then move on to other calculations like determining the object's acceleration.

Acceleration Calculation

Once you've found the net force, determining acceleration becomes straightforward with the help of Newton's Second Law. This law elegantly relates force, mass, and acceleration through the equation:

• Newton's Second Law states: \( F = ma \).
• Here, \( F \) is the net force, \( m \) is the mass of the object, and \( a \) is the acceleration.

Rearranging for acceleration gives us: \( a = \frac{F}{m} \).

For the toy plane:

• Net force \( F \) is 7 N.
• Mass \( m \) is 1.0 kg.
• Acceleration \( a \) can be calculated as \( a = \frac{7\, \text{N}}{1.0\, \text{kg}} = 7\, \text{m/s}^2 \).

This result shows how the net force and mass combine to generate an acceleration of 7 m/s², explaining how quickly the toy plane speeds up.

Physics Problem Solving

Solving physics problems effectively requires a structured approach. Start by clearly understanding the problem and identifying all forces.

Follow these steps to tackle similar challenges:

• Identify Known and Unknown Values: Determine what is given and what needs calculation. Here, the known values were the forces and the mass.
• Calculate Net Force: Sum up all forces, considering their directions, to find the net force acting on the object.
• Use Relevant Equations: Apply Newton's laws or any necessary physics formulas to find unknowns like acceleration.
• Perform Calculations: Substitute known values into the equations and solve step by step.
• Check Your Work: Verify the solution makes sense physically and mathematically.

By consistently applying these steps, you can confidently solve physics problems related to dynamics, ensuring you understand both the process and the principles behind it.