Problem 49 - College Physics

Question

A 2000 -kg car moving east at \(10.0 \mathrm{~m} / \mathrm{s}\) collides with a \(3000-k g\) car moving nouth. The cars stick together and move as a unit after the collision, at an angle of \(40.0^{\circ}\) north of east and a speed of \(5.22 \mathrm{~m} / \mathrm{s}\). Find the speed of the \(3000-\mathrm{kg}\) car before the collision.

Step-by-Step Solution

Answer

The speed of the 3000-kg car before the collision was 5.58 m/s north.

Step 1: Analyze the problem and break it down

First, note down the given data: mass of Car 1 (m1) = 2000 kg, velocity of Car 1 (u1) = 10 m/s east, mass of Car 2 (m2) = 3000 kg, angle of motion after collision (θ) = 40 degrees, and velocity after collision (v) = 5.22 m/s. We will use Newton's second law, breaking it into components to find the velocity of Car 2. The challenge in this question is to find the velocity of the second car before impact, let's name it as \(u2\).

Step 2: Apply the Law of Conservation of Momentum

Express the principle of conservation of momentum for the system. The law of conservation of momentum states that the total momentum before the collision should be equal to the total momentum after the collision. We first calculate the momentum for the system before collision in vector form. For Car 1: \(P1_i = m1 * u1 = 2000 kg * 10 m/s = 20000 kg*m/s\) (east), For Car 2: \(P2_i = m2 * u2 = 3000 kg * u2\) (north).

Step 3: Calculate the momentum after collision

The net momentum after the collision will be in the north-east direction: \(P_f = (m1 + m2) * v = (2000 kg + 3000 kg) * 5.22 m/s = 26100 kg*m/s\). This momentum needs to be broken down into its eastward and northward components. \(P_xf = P_f * cos(θ) = 26100 kg*m/s * cos(40) = 19964 kg*m/s\) (east) & \(P_yf = P_f * sin(θ) = 26100 kg*m/s * sin(40) = 16729 kg*m/s\) (north).

Step 4: Equate the momentum before and after collision

According to the law of conservation of momentum, the net momentum before collision should equal the net momentum after collision. For the x (east) direction: \(P1_i = P_xf\), thus, \(20000 kg*m/s = 19964 kg*m/s\). For the y (north) direction: \(P2_i = P_yf\), which simplifies to \(3000 kg * u2 = 16729 kg*m/s\).

Step 5: Solve for unknown velocity

To find the speed of the 3000-kg car before the collision, we just manipulate the formula of the northward momentum: \(u2 = P_yf / m2 = 16729 kg*m/s / 3000 kg = 5.58 m/s\). So, the speed of the 3000-kg car before the collision was 5.58 m/s north.

Key Concepts

Newton's Second Law

Newton's Second Law is a cornerstone of classical mechanics. It connects the net force acting on an object to its mass and acceleration by stating: \( F = ma \), where \( F \) represents force, \( m \) is mass, and \( a \) is acceleration. This fundamental principle explains how forces cause changes in motion.

In our exercise, it is crucial to extend Newton's Second Law into understanding how momentum works in collisions. Momentum is the product of an object’s mass and velocity. When a force acts over time, it changes the momentum of an object. This change in momentum is known as impulse, given by \( F\Delta t = m\Delta v \).

• Force \( F \) applied on an object over time \( \Delta t \) results in a change in velocity \( \Delta v \).
• This forms the basis of analyzing changes in motion due to collisions, as seen when one car collides with another.

Understanding momentum in terms of forces provides insight into how the cars' velocities changed after their collision. Newton's Second Law lays groundwork for comprehending the motion and resulting changes after a collision, which is pivotal when dealing with multi-directional impact problems like the one in our exercise.

Collision Physics

Collision physics revolves around two or more bodies interacting forcefully with each other, often resulting in exchanges of energy and momentum. In physics, collisions are typically classified into elastic and inelastic collisions.

• Elastic collisions are where both kinetic energy and momentum are conserved.
• Inelastic collisions, like the one in our exercise, conserve momentum but not kinetic energy as some energy gets converted into heat or deformation.

Our problem involves a perfectly inelastic collision, since the cars stick together post-collision, moving as a single unit. This is a common condition in real-life accidents because cars often undergo deformations. Therefore, it doesn't conserve kinetic energy, but the total momenta of the cars are still conserved.

The conservation of momentum is central to solving these problems, given by \[ m_1u_1 + m_2u_2 = (m_1 + m_2)v \]. Insights from collision physics allow us to break down the cars’ initial and final momenta to deduce velocities in various directions, as vehicles tend to move in different directions upon collision, as demonstrated in the given scenario.

Vector Components

Vectors describe quantities that have both magnitude and direction, like velocity and momentum in physics. The use of vector components is crucial for accurately analyzing problems where objects move in more than one dimension.

In our exercise, the car movements and collisions involve interactions in both east and north directions, which must be handled using vector components. To do this, we break down any vector into perpendicular components based on the angle it forms with a reference direction (usually the x-axis).

• For eastward and northward momentum, we use trigonometric functions: cosine for the eastward and sine for the northward components.
• This allows us to calculate momentum before and after collision accurately in two different directions, separately.

Vector components ensure calculations account for all forces acting on the system, giving us a clearer picture of the conservation laws at play. Correctly resolving vectors into components is necessary for solving multi-dimensional collision problems accurately, helping to find unknown velocities as displayed in our exercise solution.