Question

If the forces that two interacting objects exert on each other are always exactly equal in magnitude and opposite in direction, how is it possible for an object to accelerate?

Short answer

Answer: An object can accelerate even if the forces between two interacting objects are equal in magnitude and opposite in direction due to the net force acting on the object. Although the forces between the two objects are equal and opposite according to Newton's third law, an object will still accelerate if there is a non-zero net force acting on it. It is the net force that determines an object's acceleration, not the individual forces within the system.

Step-by-step solution

Understand Newton's Third Law

Newton's third law of motion states that for every action, there is an equal and opposite reaction. In other words, when two objects interact, they exert equal and opposite forces on each other. This law is important for understanding how forces work in various situations.

Define Acceleration

Acceleration is defined as the rate at which an object's velocity changes over time, and it is expressed as a vector quantity. Mathematically, acceleration (a) is given by the formula: a = (final velocity - initial velocity) / time Acceleration can be caused by a change in an object's speed, direction, or both. To have acceleration, an object must have a net force acting upon it.

Explain the Concept of Net Force

A net force is the vector sum of all the forces acting on an object. If the forces acting on an object are not balanced (i.e., equal in magnitude and opposite in direction), then there will be a net force, which in turn can cause acceleration. In other words, an object will only accelerate if there is a non-zero net force acting upon it.

Apply Newton's Third Law to Acceleration

Given that the forces between two interacting objects are equal and opposite, it may seem counterintuitive that an object can accelerate. However, it is important to remember that the forces being discussed are the forces between two different objects, not just one object. For example, consider a person pushing a box on a frictionless surface. The person exerts a force on the box, and the box exerts an equal and opposite force on the person. According to Newton's third law, these forces between the person and the box are equal in magnitude and opposite in direction. However, the person's force on the box causes the box to accelerate, while the box's force on the person does not necessarily cause the person to accelerate (depending on other forces acting on the person, such as friction). In this scenario, the net force acting on the box (i.e., the person's force minus the box's force on the person) is non-zero, which leads to acceleration. Therefore, even though the forces between the two objects are equal and opposite, the net force acting on any single object within the system can still cause acceleration.

Conclusion

An object can accelerate even if the forces between two interacting objects are equal in magnitude and opposite in direction per Newton's third law. This is because an object's acceleration is determined by the net force acting upon it, which can still be non-zero even when the forces within a system are equal and opposite.

Key concepts

Acceleration

Acceleration is a fundamental concept in physics that describes how quickly an object changes its velocity. It's not just about speed, but also the direction in which an object is moving. When an object speeds up, slows down, or changes direction, it is said to be accelerating.

Technically, acceleration is defined as the change in velocity per unit of time. The formula for acceleration \textbf{(a)} is: \[ a = \frac{\text{final velocity} - \text{initial velocity}}{\text{time}} \] This formula shows that if there is any change in velocity, whether an increase or a decrease, there will be acceleration.

It's worth noting that acceleration is a vector quantity, meaning it has both magnitude and direction. The continued motion or change in motion of an object doesn't just depend on the magnitude of the acceleration but also on the way the object is directed towards. A car turning a corner at a constant speed is still accelerating because the direction of its motion is changing.

Net Force

Net force is the total force that causes an object to move or change its velocity. It's a vector quantity as well, summing up all the individual forces acting on an object in different directions. When you add up these forces, if they are not balanced, the object will experience a net force, leading to acceleration.

To better understand net force, imagine a tug-of-war game. If two teams pull with equal force, the net force is zero, and neither team moves. However, if one team exerts a greater force, there's a net force in the direction of that team, resulting in movement or acceleration towards them. In physics, the net force \textbf{(F_{net})} is the total of all forces acting on an object, and it's calculated as: \[ F_{net} = \sum F_i \] where \( F_i \) represents individual forces. Remember, when the net force is greater than zero, an object will accelerate according to Newton's second law, \( F = ma \), where \( F \) is the net force, \( m \) is mass, and \( a \) is acceleration.

Vector Quantity

In physics, a vector quantity is one that includes both magnitude, which tells us the size or quantity, and direction, telling us the orientation of that magnitude in space. Examples of vector quantities include velocity, force, and acceleration.

Understanding vector quantities is essential in analyzing physical phenomena. For instance, when two forces act upon an object in opposite directions, their vector nature allows us to calculate the net force as the difference between those two forces.

Say you have two vectors, \( \vec{A} \) and \( \vec{B} \) — imagine these as arrows pointing in specific directions. If \( \vec{A} \) represents a force of 5 Newtons to the right, and \( \vec{B} \) represents a force of 3 Newtons to the left, the net force \( \vec{F}_{net} \) would be \( \vec{A} + (-\vec{B}) \) or simply 2 Newtons to the right. The right direction is where the object will accelerate due to the net force. This calculation shows how the vector nature of quantities like force and acceleration governs the movement of objects in physics.