Problem 80 - University Physics with Modern Physics

Question

Automobile Accident Analysis. You are called as an expert witness to analyze the following auto accident: Car \(B,\) of mass \(1900 \mathrm{kg},\) was stopped at a red light when it was hit from behind by car \(A,\) of mass 1500 \(\mathrm{kg}\) . The cars locked bumpers during the collision and slid to a stop with brakes locked on all wheels. Measurements of the skid marks left by the tires showed them to be 7.15 \(\mathrm{m}\) long. The coefficient of kinetic friction between the tires and the road was 0.65 . (a) What was the speed of car A just before the collision? (b) If the speed limit was 35 \(\mathrm{mph}\) , was car \(A\) speeding, and if so, by how many miles per hour was it exceeding the speed limit?

Step-by-Step Solution

Answer

Car A was traveling at approximately 40.3 mph before the collision and was exceeding the speed limit by about 5.3 mph.

Step 1: Understand the Formula Needed

In this problem, we need to find the speed of car A just before the collision. The principle of conservation of momentum and the concept of work-energy related to friction will be applied. We must understand:1. Conservation of momentum: \( m_A v_{A\text{before}} = (m_A + m_B) v_{\text{after}} \)2. Work done by friction to stop both cars: \( \text{work} = \text{friction force} \times \text{distance} = \frac{1}{2}(m_A + m_B)v_{\text{after}}^2 \)

Step 2: Calculate the Frictional Force

The cars come to a stop due to the frictional force. To find the frictional force, use:\[ F_{\text{friction}} = \mu_k \times m_{\text{combined}} \times g \]where \( \mu_k = 0.65 \) is the coefficient of kinetic friction, \( m_{\text{combined}} = m_A + m_B = 3400\,\text{kg} \) is the total mass, and \( g = 9.8\,\text{m/s}^2 \) is gravity.

Step 3: Solve for the Velocity after Collision

Using the work-energy principle and the skid distance, \[ F_{\text{friction}} \times 7.15 = \frac{1}{2} \times m_{\text{combined}} \times v_{\text{after}}^2 \]Substitute the values to find \( v_{\text{after}} \) (velocity of the combined mass right after the collision).

Step 4: Use Conservation of Momentum

Now apply the conservation of momentum:\[ m_A v_{A\text{before}} = m_{\text{combined}} v_{\text{after}} \]Solve for car A's velocity before collision \( v_{A\text{before}} \).

Step 5: Compare Speed to Speed Limit

Convert the speed from m/s to mph and compare it to the speed limit of 35 mph to determine how many mph car A was exceeding the speed limit.

Key Concepts

Kinetic Friction in Physics

In the scenario of a car accident where vehicles come to a halt after a collision, kinetic friction plays a crucial role in bringing the cars to a stop. When two vehicles like cars A and B collide and skid together, kinetic friction is what eventually stops them. The force of friction depends on several key factors:

• The coefficient of kinetic friction (\(\mu_k\)): This dimensionless number represents the ratio of the frictional force between the tires and the road surface to the normal force (weight) acting on the car. In our case, \(\mu_k = 0.65\).
• The combined mass of the vehicles: After the collision, the total mass \(m_{\text{combined}}\) is the sum of car A's and car B's masses, resulting in \(m_{\text{combined}} = 3400 \text{ kg}\).
• The gravitational acceleration (\(g\)): Usually taken as \(9.8 \, \text{m/s}^2\), it acts downward, affecting the normal force which equals the weight of the vehicles.

Given these factors, the frictional force \(F_{\text{friction}}\) that resists the motion can be calculated by the formula \(F_{\text{friction}} = \mu_k \times m_{\text{combined}} \times g\). This frictional force does work over the distance of the skid marks to bring the cars to a stop.

Work-Energy Principle

The work-energy principle is instrumental in understanding how the energy associated with an object's motion changes due to forces like friction. In our collision scenario, once car A hits car B, the kinetic energy of the cars is gradually transformed into thermal energy due to friction, effectively halting their motion.
This principle is applied using the formula:

• Work done by friction: \( \text{work} = F_{\text{friction}} \times \text{distance} \)
• The change in kinetic energy: \( \frac{1}{2} m_{\text{combined}} v_{\text{after}}^2\), where \(v_{\text{after}}\) is the velocity right after the collision but before stopping.

By equating the work done by friction to the change in kinetic energy, \( F_{\text{friction}} \times 7.15 = \frac{1}{2} \times m_{\text{combined}} \times v_{\text{after}}^2 \), we calculate the velocity of the combined mass immediately after the collision. This approach shows how energy conservation principles help understand and predict post-collision speeds.

Velocity Calculation from Skid Marks

In accident analysis, determining pre-collision speeds is crucial. Here, skid marks provide critical evidence for backtracking to the speed of car A right before the collision. By inspecting the length of these marks, we can apply conservation laws and friction calculations to estimate this speed:

• Use conservation of momentum initially: \( m_A v_{A\text{before}} = m_{\text{combined}} v_{\text{after}} \)
• From the work-energy principle, derive \( v_{\text{after}} \).
Additionally, comparing this computed speed of car A with the speed limit reveals if car A was speeding precisely.
The steps involved focus on translating physical evidence (skid marks) into quantitative data like velocity, by bridging different physics concepts – from momentum conservation to energy principles. Ultimately, converting \(\text{m/s}\) to \(\text{mph}\) enables easy comparison with speed limits.