Question

An SUV of mass 3250 kg has a head-on collision with a \(1250-\mathrm{kg}\) subcompact. Identify all the statements that are incorrect. a) The SUV exerts a larger force on the subcompact than the subcompact exerts on the SUV. b) The subcompact exerts a larger force on the SUV than the SUV exerts on the subcompact. c) The subcompact experiences a larger acceleration than the SUV. d) The SUV experiences a larger acceleration that the subcompact.

Short answer

a) The SUV exerts a larger force on the subcompact than the subcompact exerts on the SUV. b) The subcompact exerts a larger force on the SUV than the SUV exerts on the subcompact. d) The SUV experiences a larger acceleration than the subcompact. Answer: a, b, d

Step-by-step solution

Analyze the forces in the collision

In a head-on collision, the forces acting on both the SUV and the subcompact car will be according to Newton's third law of motion, which states that for every action, there is an equal and opposite reaction. This means that the force exerted by the SUV on the subcompact car will be equal and opposite to the force exerted by the subcompact car on the SUV. Now let's analyze the given statements.

Analyze statement a

Statement a says that the SUV exerts a larger force on the subcompact than the subcompact exerts on the SUV. However, as we discussed in step 1, the forces exerted by both vehicles on each other are equal and opposite according to Newton's third law of motion. Therefore, statement a is incorrect.

Analyze statement b

Statement b says that the subcompact exerts a larger force on the SUV than the SUV exerts on the subcompact. This statement is also false for the same reason as in step 2; the forces exerted by both vehicles on each other are equal and opposite as per Newton's third law of motion.

Analyze statement c

Statement c says that the subcompact experiences a larger acceleration than the SUV. To verify this statement, we will use Newton's second law of motion, which states that F = ma, where F is the force acting on an object, m is the mass of the object, and a is the acceleration of the object. Since the forces on both vehicles are equal, and the subcompact car has a smaller mass than the SUV, it will experience a greater acceleration as per the equation. Therefore, statement c is correct.

Analyze statement d

Statement d says that the SUV experiences a larger acceleration than the subcompact. This statement contradicts statement c, which we determined to be correct in step 4. Since the subcompact has a smaller mass and experiences the same force as the SUV, it will have a greater acceleration than the SUV. Therefore, statement d is incorrect. In conclusion, the incorrect statements are: a) The SUV exerts a larger force on the subcompact than the subcompact exerts on the SUV. b) The subcompact exerts a larger force on the SUV than the SUV exerts on the subcompact. d) The SUV experiences a larger acceleration than the subcompact.

Key concepts

Head-On Collision Physics

Understanding the dynamics of a head-on collision involves several principles of physics, particularly the concepts laid out by Sir Isaac Newton. A head-on collision is an event where two vehicles or objects collide front-first. During this intense interaction, the concept of impulse, which is the product of force and the time for which it is applied, plays a crucial role.

Impulse is directly related to the momentum change experienced by the colliding bodies. Remember, momentum is the product of an object's mass and velocity, and it is a conserved quantity in a closed system, meaning the total momentum before and after a collision remains constant in the absence of external forces. In a head-on collision, each vehicle experiences an impulse that leads to momentum changes that are equal in magnitude but opposite in direction. This is where Newton's laws come into the picture, particularly his third law which discusses action-reaction pairs.

Newton's Second Law of Motion

Newton's second law of motion is pivotal to understanding how forces affect motion. It can be succinctly expressed with the formula F = ma, where F stands for force, m for mass, and a for acceleration. This law tells us that the acceleration of an object is directly proportional to the net force acting on it and inversely proportional to its mass. The direction of acceleration is in the direction of the applied net force.

When applying this to the SUV and subcompact collision problem, it becomes clear why the subcompact will experience a greater acceleration. The SUV and subcompact car experience the same magnitude of force during the collision, but because the subcompact is lighter, it will have a higher acceleration. This illustrates the inverse relationship between mass and acceleration when the force is constant.

Another critical point to note is that the second law provides a quantitative measure of the forces involved in any motion, which is incredibly useful when predicting the result of force application in various physical situations.

Newton's Third Law of Motion

Newton's third law of motion states that for every action, there is an equal and opposite reaction. This means that whenever an object exerts a force on a second object, the second object exerts an equal and opposite force on the first. This law highlights the nature of forces as interactions that cannot occur in isolation but rather always in pairs.

In the context of the head-on collision between the SUV and the subcompact car, this law implies that the force exerted by the SUV on the subcompact is equal in magnitude and opposite in direction to the force that the subcompact exerts on the SUV. Thus, it is incorrect to claim that one car exerts a larger force on the other; they exert equal and opposite forces on each other, regardless of their masses or the aftermath of the collision.

Force and Acceleration Relationship

The relationship between force and acceleration is central to understanding how objects move and interact. This relationship, dictated by Newton's second law of motion, tells us that the acceleration of an object is directly proportional to the net force acting upon it when mass is held constant. Put another way, if you apply a greater force to an object, its acceleration increases correspondingly if the mass of the object doesn't change.

In the case of a collision, this relationship helps explain the differing accelerations of the SUV and subcompact car after the impact. The force experienced by both is identical due to Newton's third law, but the SUV, which has more mass, will have less acceleration compared to the lighter subcompact car. This illustrates the inverse relationship between mass and acceleration and is vital in predicting the behavior of objects under the influence of various forces.