Problem 39 - University Physics with Modern Physics

Question

Two cars, one a compact with mass 1200 \(\mathrm{kg}\) and the other a large gas-guzzler with mass \(3000 \mathrm{kg},\) collide head-on at typical freeway speeds. (a) Which car has a greater magnitude of momentum change? Which car has a greater velocity change? (b) If the larger car changes its velocity by \(\Delta v\) , calculate the change in the velocity of the small car in terms of \(\Delta v .\) (c) Which car's occupants would you expect to sustain greater injuries? Explain.

Step-by-Step Solution

Answer

Both cars have equal momentum change; compact car has greater velocity change. The compact car's velocity change is 2.5 times greater. Compact car's occupants likely sustain greater injuries.

Step 1: Momentum Change Analysis

In a head-on collision, according to the law of conservation of momentum, the total momentum before collision equals the total momentum after collision. During the collision, the momentum lost by one car is gained by the other. Therefore, both cars experience the same magnitude of momentum change, even though their masses are different.

Step 2: Velocity Change Analysis

The change in velocity is given by the formula \( \Delta v = \frac{\Delta p}{m} \), where \( \Delta p \) is the change in momentum and \( m \) is the mass. Since momentum change \( \Delta p \) is the same for both cars, the smaller mass (the compact car) will result in a greater velocity change. Thus, the compact car will have a greater velocity change.

Step 3: Calculate Velocity Change of Small Car

Let the velocity change of the large car be \( \Delta v \). The large car's momentum change is \( m_{large} \cdot \Delta v = 3000\, \mathrm{kg} \times \Delta v \).For the compact car, momentum change is equal to the large car's momentum change, so:\[ m_{small} \cdot \Delta v_{small} = 3000\, \mathrm{kg} \times \Delta v \]Solving for \( \Delta v_{small} \), we get:\[ \Delta v_{small} = \frac{3000}{1200} \times \Delta v = 2.5 \cdot \Delta v \]Therefore, the change in velocity of the compact car is 2.5 times the change in velocity of the large car.

Step 4: Evaluate Occupant Injuries

In a collision, the occupants of the car with greater velocity change experience more rapid deceleration, which typically leads to greater force applied to their bodies (as force is tied to acceleration through Newton's second law \( F = m \cdot a \)). Therefore, the occupants of the compact car, which undergoes a greater velocity change, would likely sustain greater injuries.

Key Concepts

Conservation of Momentum

When two objects collide, their total momentum before the collision is the same as their total momentum after the collision. This is known as the conservation of momentum principle.

In the case of our two cars colliding, the compact car and the gas-guzzler, the rule of conservation of momentum tells us that the total momentum they share doesn't change despite the crash.

Here's how it works:

• The compact car, with a mass of 1200 kg, and the large car, with a mass of 3000 kg, both have momentum dependent on their velocities just before the crash.
• During the collision, the momentum one car loses equals the momentum the other car gains.
• So, their momentum changes are equal and opposite, leading to no net momentum change in the system.

By maintaining the overall momentum, we see that while the collisions appear chaotic, the principle of momentum conservation brings an underlying order.

Velocity Change

Velocity change is a crucial part we must consider when discussing collisions. Even though the cars have identical momentum changes due to conservation, their masses influence how much their velocities change.

The formula used here is \( \Delta v = \frac{\Delta p}{m} \), where \( \Delta p \) is the change in momentum and \( m \) is the mass of the car.

• Both cars undergo the same change in momentum. However, the compact car, because of its lighter mass, experiences a larger velocity change.
• If the larger car's velocity changes by \( \Delta v \), the formula shows that the compact car’s velocity changes by 2.5 times that of the large car.
• This occurs because a smaller mass means a more significant velocity change for a given amount of momentum change.

Thus, velocity change not only reflects the impact of mass but also highlights its effect on occupants' experiences in a collision.

Force and Acceleration

In a collision, understanding force and acceleration is integral to predicting the effects on a vehicle's occupants. When velocity changes, acceleration occurs, and this is what ultimately causes force.

Using Newton's second law, \( F = m \cdot a \), we can comprehend that:

• The acceleration is directly related to the change in velocity; more dramatic changes in velocity lead to higher accelerations.
• For the compact car, which undergoes a greater velocity change, the resulting acceleration is higher.
• This acceleration translates into a larger force experienced by the car's occupants.

Rapid deceleration or a significant reduction in speed means the occupants feel a stronger impact force.

Therefore, even though both cars undergo the same change in momentum, the compact car’s greater change in velocity and acceleration levels suggests its occupants might sustain more severe injuries. This underscores the importance of vehicle safety features designed to mitigate such forces.