Problem 83 - College Physics

Question

. Accident analysis. A 1500 \(\mathrm{kg}\) sedan goes through a wide intersection traveling from north to south when it is hit by a 2200 kg SUV traveling from east to west. The two cars become enmeshed due to the impact and slide as one there after. On-the-scene measurements show that the coefficient of kinetic friction between the tires of these cars and the pave- ment is \(0.75,\) and the cars slide to a halt at a point 5.39 \(\mathrm{m}\) west and 6.43 \(\mathrm{m}\) south of the impact point. How fast was each car traveling just before the collision?

Step-by-Step Solution

Answer

The sedan was traveling at approximately 18.2 m/s and the SUV at 12.3 m/s before the collision.

Step 1: Analyze Momentum Conservation

The accident results in an inelastic collision where both vehicles move together after the collision. By conserving linear momentum, the total initial momentum in each direction equals the combined momentum immediately after the collision. Consider the momentum equations:- In the north-south (j) direction:\[ m_{ ext{sedan}} imes v_{ ext{sedan}} = (m_{ ext{sedan}} + m_{ ext{SUV}}) imes v_{ ext{combined}} imes rac{6.43}{ ext{Total Distance}} \]- In the east-west (i) direction:\[ m_{ ext{SUV}} imes v_{ ext{SUV}} = (m_{ ext{sedan}} + m_{ ext{SUV}}) imes v_{ ext{combined}} imes rac{5.39}{ ext{Total Distance}} \]

Step 2: Calculate the Total Distance

The cars slide to a halt - Distance east-west: 5.39 m- Distance north-south: 6.43 mUse the Pythagorean theorem to determine the total distance traveled by the cars after the collision:\[ \text{Total Distance} = \sqrt{(5.39)^2 + (6.43)^2} \approx 8.35 \, \text{m} \]

Step 3: Calculate the Velocity of Cars Post-Collision

The skid is due to kinetic friction, which allows for calculating the deceleration using:\[ f_k = \mu_k \times m_{ ext{combined}} \times g = m_{ ext{combined}} \times a \]\[ a = \mu_k \times g = 0.75 \times 9.8 \, \text{m/s}^2 = 7.35 \, \text{m/s}^2 \]Using kinematics, find initial speed after collision:\[ v_{ ext{post}}^2 = 2a \times \text{Total Distance} \rightarrow v_{ ext{post}} = \sqrt{2 \times 7.35 \times 8.35} \approx 11.02 \, \text{m/s} \]

Step 4: Set up Equations Using Momentum Conservation

Substitute values and expressions obtained into momentum equations:- North-south direction:\[ 1500 \times v_{ ext{sedan}} = 3700 \times 11.02 \times \frac{6.43}{8.35} \]- East-west direction:\[ 2200 \times v_{ ext{SUV}} = 3700 \times 11.02 \times \frac{5.39}{8.35} \]

Step 5: Solve for Initial Velocities

Solve these linear momentum conservation equations:- For the sedan:\[ v_{ ext{sedan}} \approx \frac{3700 \times 11.02 \times \frac{6.43}{8.35}}{1500} \approx 18.2 \, \text{m/s} \]- For the SUV:\[ v_{ ext{SUV}} \approx \frac{3700 \times 11.02 \times \frac{5.39}{8.35}}{2200} \approx 12.3 \, \text{m/s} \]

Key Concepts

Momentum Conservation

When two vehicles collide and stick together, like in this accident, the principle of momentum conservation is at play. Momentum conservation states that the total momentum before the collision is equal to the total momentum after the collision. During an inelastic collision, where objects stick together, momentum is conserved, but kinetic energy is not.

• Momentum is the product of an object's mass and its velocity.
• In our scenario, the momentum must be calculated for the north-south and east-west directions separately. This is because momentum is a vector, having both magnitude and direction.
• Equations for momentum conservation are set up in both the north-south and east-west directions. This allows us to determine how fast each vehicle was traveling before the collision.

By analyzing these equations, we can find out the speeds of the sedan and SUV prior to impact.

Kinetic Friction

Kinetic friction plays a crucial role in this accident, especially in calculating how the cars come to a halt post-collision. Kinetic friction is the force that opposes the motion of two sliding surfaces. It is different from static friction, which acts on objects not yet in motion.

• This force depends on the coefficient of kinetic friction and the normal force, which in this case is the combined weight of the vehicles.
• The coefficient of 0.75 means there's considerable resistance slowing down the cars.
• We calculate the deceleration caused by kinetic friction to determine how the cars' speeds change over the distance they slide together.

Understanding kinetic friction helps us gauge how forces act on colliding vehicles and how they will behave when sliding across a surface.

Velocity Calculation

Determining the velocity of each vehicle before the accident is a key step in analyzing the collision. Using momentum conservation and other given parameters, we can calculate these initial velocities.
First, we determine the total distance traveled after the collision using the Pythagorean theorem, finding it to be approximately 8.35 m.

• Using the calculated acceleration due to kinetic friction, we find the combined velocity of the cars immediately after the collision to be approximately 11.02 m/s.
• Knowing this post-collision velocity allows us to work backward using the momentum conservation laws to calculate the initial velocities of both cars.
• We end up with a sedan velocity of approximately 18.2 m/s and an SUV velocity of about 12.3 m/s.

These calculations help reconstruct the events leading up to the accident by identifying how each vehicle moved just before impact.

Traffic Accident Analysis

Traffic accident analysis involves applying physics and mathematics to understand and reconstruct the incident. By evaluating the data from the accident:

• We can determine factors like the speed of the vehicles, the point of collision, and the forces involved.
• This analysis is crucial for assessing fault, understanding accident dynamics, and improving safety standards.
• For this scenario, we combine observations, such as the skid distance and directions, with laws of physics like momentum conservation and friction.

The insights gained from such detailed analyses not only point to what happened in a specific accident but also allow for preventive measures to avoid future occurrences.