Problem 14 - College Physics

Question

A \(750 \mathrm{~kg}\) car is stalled on an icy road during a snowstorm. A \(1000 \mathrm{~kg}\) car traveling eastbound at \(10 \mathrm{~m} / \mathrm{s}\) collides with the rear of the stalled car. After being hit, the \(750 \mathrm{~kg}\) car slides on the ice at \(4 \mathrm{~m} / \mathrm{s}\) in a direction \(30^{\circ}\) north of east. (a) What are the magnitude and direction of the velocity of the \(1000 \mathrm{~kg}\) car after the collision? (b) Calculate the ratio of the kinetic energy of the two cars just after the collision to that just before the collision. (You may ignore the effects of friction during the collision.)

Step-by-Step Solution

Answer

(a) The final velocity of the 1000 kg car is approximately 8.83 m/s at 9.76° south of east. (b) The kinetic energy ratio after to before the collision is 0.8996.

Step 1: Determine Initial Momentum

To solve part (a), we begin by applying the conservation of momentum. The total momentum before the collision must equal the total momentum after the collision. Initially, only the 1000 kg car is moving, so the initial momentum is given by: \[ p_{i} = m_{1} \cdot v_{1i} + m_{2} \cdot v_{2i} \] where \( m_{1} = 1000\, \text{kg} \), \( v_{1i} = 10\, \text{m/s} \), \( m_{2} = 750\, \text{kg} \), and \( v_{2i} = 0\, \text{m/s} \) (as the car is stalled). Substitute these values in: \[ p_{i} = 1000 \cdot 10 + 750 \cdot 0 = 10000 \, \text{kg}\cdot\text{m/s} \] in the eastward direction.

Step 2: Resolve Final Velocities into Components

After the collision, the 750 kg car moves at \( 4 \, \text{m/s} \) at \( 30^{\circ} \) north of east. We resolve this into x (east) and y (north) components using trigonometry: \[ v_{2xf} = 4 \cos(30^{\circ}) = 4 \times \frac{\sqrt{3}}{2} = 2\sqrt{3} \, \text{m/s} \] \[ v_{2yf} = 4 \sin(30^{\circ}) = 4 \times \frac{1}{2} = 2 \, \text{m/s} \] The 1000 kg car's final velocity is \( v_{1f} = \begin{pmatrix} v_{1xf} \ v_{1yf} \end{pmatrix} \). We will find these components next.

Step 3: Apply Conservation of Momentum Along X-axis

By momentum conservation along the x-axis: \[ m_{1} \cdot v_{1xi} = m_{1} \cdot v_{1xf} + m_{2} \cdot v_{2xf} \] \[ 1000 \cdot 10 = 1000 \cdot v_{1xf} + 750 \cdot 2\sqrt{3} \] \[ 10000 = 1000 \cdot v_{1xf} + 1299.04 \] Solving for \( v_{1xf} \): \[ v_{1xf} = \frac{10000 - 1299.04}{1000} = 8.701 \, \text{m/s} \]

Step 4: Apply Conservation of Momentum Along Y-axis

By momentum conservation along the y-axis: \[ 0 = m_{1} \cdot v_{1yf} + m_{2} \cdot v_{2yf} \] \[ 0 = 1000 \cdot v_{1yf} + 750 \cdot 2 \] \[ 1000 \cdot v_{1yf} = -1500 \] Solving for \( v_{1yf} \): \[ v_{1yf} = -1.5 \, \text{m/s} \]

Step 5: Calculate Magnitude and Direction of Final Velocity

The final velocity of the 1000 kg car can be determined using its x and y components: \( v_{1xf} = 8.701 \, \text{m/s} \), \( v_{1yf} = -1.5 \, \text{m/s} \). The magnitude \( v_{1f} \) is calculated as: \[ v_{1f} = \sqrt{(v_{1xf})^2 + (v_{1yf})^2} = \sqrt{(8.701)^2 + (-1.5)^2} \approx 8.83 \, \text{m/s} \] The direction \( \theta \) (south of east) is given by: \[ \theta = \tan^{-1}\left(\frac{-1.5}{8.701}\right) \approx -9.76^{\circ} \] Thus, the velocity is \( 8.83 \, \text{m/s} \) at \( 9.76^{\circ} \) south of east.

Step 6: Calculate Initial and Final Kinetic Energy

For part (b), calculate the kinetic energy before and after collision. The initial kinetic energy is: \[ KE_{i} = \frac{1}{2} m_{1} v_{1i}^2 = \frac{1}{2} \times 1000 \times (10)^2 = 50000 \, \text{J} \] The final kinetic energies are: \[ KE_{1f} = \frac{1}{2} m_{1} v_{1f}^2 = \frac{1}{2} \times 1000 \times (8.83)^2 \approx 38982.45 \, \text{J} \] \[ KE_{2f} = \frac{1}{2} m_{2} v_{2f}^2 = \frac{1}{2} \times 750 \times (4)^2 = 6000 \, \text{J} \] Total \( KE_{f} = 38982.45 + 6000 = 44982.45 \, \text{J} \)

Step 7: Calculate Kinetic Energy Ratio

The ratio of the total kinetic energy after to before the collision is: \[ \text{Ratio} = \frac{KE_{f}}{KE_{i}} = \frac{44982.45}{50000} \approx 0.8996 \] This indicates the kinetic energy was reduced to approximately 89.96% of its original value due to the inelastic collision.

Key Concepts

Elastic and Inelastic Collisions

Collisions are events where two or more objects exert forces on each other for a brief period of time. In the study of collisions, it's essential to understand the differences between elastic and inelastic collisions.
In an **elastic collision**, both momentum and kinetic energy are conserved. This means that the total momentum and the total kinetic energy of the colliding objects remain unchanged after the collision. An example of an elastic collision might be two billiard balls hitting each other where they bounce off retaining their speed and energy.
In contrast, **inelastic collisions** involve the conservation of momentum but not kinetic energy. This implies that while the total momentum of the system remains unchanged, some of the kinetic energy is lost to other forms of energy like heat or sound. In the given exercise, when the two cars collide and don't bounce off each other perfectly, it exemplifies an inelastic collision. This is evident by the fact that the kinetic energy after the collision is less than before the collision, reflecting the transformation of energy into non-mechanical forms.

Kinetic Energy

Kinetic energy is the energy that an object possesses due to its motion. It can be calculated using the formula: \[ KE = \frac{1}{2} m v^2 \]where \( m \) is the mass of the object and \( v \) is its velocity.
In the context of the exercise, calculating the kinetic energy at different stages of the collision helps us to understand what happens during the impact.
Initially, the 1000 kg car had a kinetic energy of 50,000 Joules, as it was the only moving car. After the collision, the total kinetic energy decreases to 44,982.45 Joules, illustrating that some energy has been dissipated. This change substantiates the nature of inelastic collisions. It's important to note that while the system loses kinetic energy, it is transformed into other forms of energy, not lost entirely. Understanding how kinetic energy changes in collisions can provide insights into the behavior of colliding bodies and the systems involved.

Momentum Components

Momentum is a vector quantity, meaning it has both magnitude and direction. It is defined by the formula:\[ p = m v \]Given that momentum is directional, it's essential to extrapolate its components especially when dealing with interactions in two dimensions.
In the exercise, after the collision, the movement of the cars occurs along two dimensions: east and north. We can resolve the velocity of the stalled car into two directional components using trigonometry, yielding east and north components. This analysis allows us to better understand the changes in momentum.
Conservation of momentum in each directional component separately, both along the x (east-west) and y (north-south) axes, ensures calculations can precisely predict the final velocities of the objects involved. By breaking down velocities into components, physicists can meticulously characterize the final state of objects post-collision, aiding in detailed and accurate analysis of these dynamic systems.