A small car of mass $380$ kg is pushing a large truck of mass $900$ kg due east on a level road. The car exerts a horizontal force of $1600$ N on the truck. What is the magnitude of the force that the truck exerts on the car?

Verified step by step guidance

1. Step 1: Begin by identifying the forces acting on the car and the truck. The car exerts a force of 1600 N on the truck, and according to Newton's Third Law, the truck exerts an equal and opposite force on the car. This is the force we are solving for.
2. Step 2: Recognize that the car and truck are moving together as a system. Calculate the acceleration of the system using Newton's Second Law: \( F_{net} = (m_{car} + m_{truck}) \cdot a \), where \( F_{net} \) is the net force exerted by the car (1600 N), \( m_{car} \) is the mass of the car (380 kg), \( m_{truck} \) is the mass of the truck (900 kg), and \( a \) is the acceleration.
3. Step 3: Rearrange the formula to solve for acceleration \( a \): \( a = \frac{F_{net}}{m_{car} + m_{truck}} \). Substitute the values for \( F_{net} \), \( m_{car} \), and \( m_{truck} \) into the equation.
4. Step 4: Once the acceleration \( a \) is determined, calculate the force exerted by the truck on the car. Use Newton's Second Law for the car: \( F_{car} = m_{car} \cdot a \). This force is the reaction force exerted by the truck on the car.
5. Step 5: Apply Newton's Third Law to confirm that the magnitude of the force exerted by the truck on the car is equal to the force exerted by the car on the truck (1600 N), but in the opposite direction.

Key Concepts

Newton's Third Law of Motion

Newton's Third Law states that for every action, there is an equal and opposite reaction. This means that if one object exerts a force on a second object, the second object exerts a force of equal magnitude but in the opposite direction on the first object. In this scenario, the force exerted by the car on the truck will be matched by the force exerted by the truck on the car.

Force and Mass

Force is defined as the interaction that causes an object to change its velocity, and it is calculated using Newton's second law, F = ma, where F is force, m is mass, and a is acceleration. In this problem, understanding the masses of both the car and the truck is essential to analyze the forces involved and how they interact with each other.

Net Force and Acceleration

The net force acting on an object is the vector sum of all the forces acting on it, which determines its acceleration according to Newton's second law. In this case, the net force on the truck is influenced by the force exerted by the car, and understanding how these forces interact helps in determining the acceleration of both vehicles and the forces they exert on each other.