Problem 73 - 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 \(2200 \mathrm{~kg}\) SUV traveling from east to west. The two cars become enmeshed due to the impact and slide as one thereafter. On-the-scene measurements show that the coefficient of kinetic friction between the tires of these cars and the pavement 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 SUV was traveling at 10 m/s, and the sedan was traveling at 17.5 m/s before the collision.

Step 1: Understand the problem

We have a collision between a sedan and an SUV. After the collision, they move together and slide to a stop. We need to find the speed of each vehicle just before the collision using the conservation of momentum.

Step 2: Calculate post-collision velocity

Because the two vehicles are locked together after the collision, we'll determine their common velocity just after the collision. To do this, we will use the work-energy principle:The work done by friction (which causes them to stop) equals the kinetic energy they had just after the collision.The work done by friction is given by:\[ \text{Work} = f_k \times \text{distance} = \mu \times m \times g \times d \]Therefore: \[ f_k = \mu \times (m_1 + m_2) \times g = 0.75 \times (1500 + 2200) \times 9.81 \]\[ \text{distance} = \sqrt{(5.39)^2 + (6.43)^2} \approx 8.363 \text{ m} \]Set the work done by friction equal to the initial kinetic energy:\[ \frac{1}{2} \times (m_1 + m_2) \times v^2 = 0.75 \times 3700 \times 9.81 \times 8.363 \]Solving for \(v\):\[ v = \sqrt{2 \times \frac{(0.75 \times 3700 \times 9.81 \times 8.363)}{3700}} \approx 9.22 \text{ m/s}\]

Step 3: Apply conservation of momentum

Initial momentum in each direction (x for east-west, y for north-south) is conserved. This yields the following equations:For the x-direction (east-west):\[ m_{\text{SUV}} \cdot v_{x_{\text{SUV}}} = (m_{\text{sedan}} + m_{\text{SUV}}) \cdot v_x\]For the y-direction (north-south):\[ m_{\text{sedan}} \cdot v_{y_{\text{sedan}}} = (m_{\text{sedan}} + m_{\text{SUV}}) \cdot v_y\]Using trigonometry, since the velocity of the enmeshed cars just after collision was 9.22 m/s and they slid 5.39 m west and 6.43 m south, we calculate their velocities in each direction:\[ v_x = 9.22 \times \frac{5.39}{8.363} \approx 5.93 \text{ m/s} \]\[ v_y = 9.22 \times \frac{6.43}{8.363} \approx 7.09 \text{ m/s} \]

Step 4: Solve for initial speeds

Using the values from Step 3:For the SUV:\[ 2200 \times v_{x_{\text{SUV}}} = 3700 \times 5.93 \]\[ v_{x_{\text{SUV}}} \approx 10 \text{ m/s} \]For the Sedan:\[ 1500 \times v_{y_{\text{sedan}}} = 3700 \times 7.09 \]\[ v_{y_{\text{sedan}}} \approx 17.5 \text{ m/s} \]

Key Concepts

Conservation of Momentum

In collision analysis, one of the most crucial principles is the conservation of momentum. It states that the total momentum of a closed system remains constant if no external forces act on it. In the case of our accident involving a sedan and an SUV, the principle of conservation of momentum applies just before and just after the collision. This means that the combined momentum of the sedan and the SUV before they collided is equal to their combined momentum immediately after collision.

When two objects collide and stick together, like in our scenario, the momentum in each direction (east-west and north-south) must be calculated separately. The initial momentum in the east-west direction (caused by the SUV) combined with the initial momentum in the north-south direction (from the sedan) should equal the momentum in those directions just after the collision. Thus, while the two vehicles became entangled, their total momentum was conserved until they came to a halt due to friction forces.

Understanding momentum conservation is fundamental in solving problems like this, as it allows us to find the unknown initial velocities of the vehicles using given final conditions.

Kinetic Friction

Kinetic friction plays a significant role in how objects move after a collision. It is the force that opposes the motion of two surfaces sliding past one another. The strength of this force is related to the nature of both contact surfaces and is quantified using the coefficient of kinetic friction. In our exercise, the contact is between the cars' tires and the pavement, with a known coefficient of 0.75.

The work done by kinetic friction after the collision is crucial to solving the problem as it is responsible for bringing the vehicles to a stop. The force due to kinetic friction can be calculated by multiplying the coefficient of kinetic friction, the combined mass of the entangled vehicles, and the gravitational force. Hence, the work done by this friction force is equal to the kinetic energy lost, which aids in finding the common velocity just after the collision occurred.

By understanding kinetic friction, we can determine how long and far the vehicles will slide until they completely stop, using this knowledge to backtrack the initial velocities of the vehicles involved.

Collision Analysis

Collision analysis involves examining events where two or more bodies exert forces on each other over a short time scale, resulting in a change in momentum. In our case, the problem focuses on an inelastic collision, where the sedan and SUV stick together after impact.

When conducting collision analysis, you must use the known masses and understand the physics principles involved like momentum conservation and energy dissipation. Post-collision velocities can give insights into the speeds of the vehicles before they hit each other. This is crucial since it sheds light on both the immediate effects of the collision and the dynamics leading up to it.

One key aspect of collision analysis is using trigonometry. For complex movements where both east-west and north-south directions are significant, trigonometric functions help in breaking down the resulting velocity vector into its components. This breakdown provides a deeper understanding of motion changes during the collision.

With proper analysis, collisions, though momentary, reveal a wealth of information about motion and forces at play.

Work-Energy Principle

The work-energy principle is a fundamental concept in physics that relates the work done by all forces acting on a particle to the change in kinetic energy of that particle. In this problem, it is used to link the work done by kinetic friction to the loss in kinetic energy of the vehicles as they come to a stop.

By calculating the work done by friction, which acts to stop the motion of the cars, you can determine the kinetic energy the cars possessed just after the collision. This is achieved by equating the work done by friction (using the frictional force and the distance moved) with the change in kinetic energy.

Applying the work-energy principle here is crucial to deducing the post-collision velocity of the vehicles when they first collide and stick together. Once you have this velocity, conservation principles can then assist in mapping back to the initial speeds of the sedan and SUV. This principle shows how energy transformations facilitate understanding motion and force relationships.