Question

Two cars collide at an intersection. Car \(A\), with a mass of 2000 kg, is going from west to east, while car \(B\), of mass 1500 kg, is going from north to south at 15 m/s. As a result, the two cars become enmeshed and move as one. As an expert witness, you inspect the scene and determine that, after the collision, the enmeshed cars moved at an angle of 65\(^\circ\) south of east from the point of impact. (a) How fast were the enmeshed cars moving just after the collision? (b) How fast was car \(A\) going just before the collision?

Short answer

The speed of the enmeshed cars was approximately 9.16 m/s, and car A was going approximately 17.32 m/s before the collision.

Step-by-step solution

Understand the Problem

We are given that two cars collide and become one mass after the collision. Car B's initial velocity is known, as well as the angle at which both cars move after the collision. We need to determine the speed of both cars combined and then find the speed of car A before the collision.

Use Conservation of Momentum for Both Axes

Since momentum is conserved, the momentum before the collision should equal the momentum after the collision. We apply this principle to the east-west and north-south components separately. The total mass of the enmeshed cars is 3500 kg (2000 kg + 1500 kg). The velocities of car A and B initially affect these momentum equations.

Create the Equations for Conservation of Momentum

In the east-west (x-axis) direction: \( mv_{Ax} = (m_A + m_B) v_f \cos(65^\circ) \)In the north-south (y-axis) direction: \( mv_{By} = (m_A + m_B) v_f \sin(65^\circ) \)Here, \( v_f \) is the speed after the collision, \( v_{Ax} \) is the velocity of car A, and \( v_{By} \) is the velocity of car B, which is 15 m/s.

Solve the North-South Momentum Equation

Substitute the given values into the north-south momentum equation: \( 1500 \times 15 = 3500 \times v_f \sin(65^\circ) \)Solving for \( v_f \sin(65^\circ) \), we calculate the component of velocity post-collision.

Solve for v_f Using North-South Equation

Solve the equation from Step 4:\( v_f \sin(65^\circ) = \frac{1500 \times 15}{3500} \)\( v_f = \frac{1500 \times 15}{3500 \times \sin(65^\circ)} \)

Solve for the Velocity of Car A Using the East-West Equation

Using the result from Step 5, substitute into the east-west equation:\( 2000 v_{Ax} = 3500 \times v_f \cos(65^\circ) \) and solve for \( v_{Ax} \).

Check Solution Consistency

Verify that both axes' conservation equations give consistent results and that momentum is conserved. Review calculations for any discrepancies.

Key concepts

Conservation of Momentum

Momentum is a fundamental concept in physics, especially when analyzing collisions. It is defined as the product of an object's mass and velocity. In a closed system, the total momentum remains constant unless acted upon by an external force. This is known as the conservation of momentum.

In the case of the two colliding cars, we assume that the system is closed and isolated, meaning no external forces influence the motion aside from the collision itself. Before the collision, each car has its own momentum:

• Car A moving east with an unknown velocity has a momentum that we need to find.
• Car B moving south with a momentum equal to its mass times its speed of 15 m/s.

The conservation law implies that the vector sum of these momenta will equal the momentum of the enmeshed cars after the collision. This allows us to write separate momentum equations in the east-west and north-south directions to solve for unknown velocities.

Inelastic Collision

An inelastic collision is where two objects collide and do not bounce off each other. Instead, they stick together and move as one mass afterward. In these cases, kinetic energy is not conserved, but momentum is. This is what happens with the two cars in the exercise: they become enmeshed and continue to move in a new direction, forming a larger mass composed of the combined masses of both cars.

Why is kinetic energy not conserved? During an inelastic collision, some kinetic energy is transformed into other forms of energy, such as thermal energy, sound, or internal energy changes in the material of the objects. In the exercise, calculating the total momentum after the collision gives us key information about the system. We can find the speed of the combined mass after the collision, because total momentum before = total momentum after, despite energy loss in other forms.

2D Motion Analysis

When objects move in two dimensions, we must analyze their motion along each axis separately. This is crucial for solving problems involving angled velocities. In our exercise, after the collision, the cars travel at an angle 65° south of east. This requires decomposing the motion into east-west (x-axis) and north-south (y-axis) components.

The analysis involves using trigonometric functions to break down velocities:

• The component along the east-west direction is calculated using the cosine of the angle.
• The component along the north-south direction uses the sine of the angle.

By solving the separate momentum equations for each directional component, we can deduce the individual velocities and angles, giving us a complete picture of the collision aftermath.