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Chapter 6: Problem 71

(a) A car traveling due east strikes a car traveling due north at an intersection, and the two move together as a unit. A property owner on the southeast corner of the intersection claims that his fence was torn down in the collision. Should he be awarded damages by the insurance company? Defend your answer. (b) Let the eastward-moving car have a mass of 1 300 kg and a speed of 30.0 km/h and the northward-moving car a mass of 1 100 kg and a speed of 20.0 km/h. Find the velocity after the collision. Are the results consistent with your answer to part (a)?

Short Answer

Expert verified
No, the owner on the southeast corner should not be awarded damages as the momentum would direct the cars towards north of east direction. The velocity after the collision is calculated to be 44 km/h, 29.5 degrees north of east, confirming that the unit of cars would not move southeast.

Step by step solution

01

Analysis of Part (a)

First, let's consider that when two cars collide and start moving as a unit, they follow the concept of 'conservation of momentum'. According to this concept, the direction in which this unit will move depends upon the combined momentum of the two cars at the time of collision. If one car is going due north and the other one is going due east, they will together move in the direction falling between north and east. In this case, the southeast corner should not be affected. Therefore in part (a), the property owner on the southeast corner should not be awarded damages by the insurance company. This is because the combined car-unit could not have moved to the southeast.
02

Calculation of velocity after collision

Applying the conservation of momentum in x and y direction, the momentum before collision is equal to the momentum after collision. The eastward (x-axis) momentum before collision equals to mass (m1) of eastward-moving car multiplied by its velocity (v1), 1300kg * 30km/h = 39000 kg km/h. The northward (y-axis) momentum before collision equals to mass (m2) of the northward-moving car multiplied by its velocity (v2), 1100kg * 20km/h = 22000 kg km/h.
03

Calculation of overall velocity and direction

To find the velocity of the two cars after collision, create a vector triangle with eastward momentum as the adjacent side and northward momentum as the opposite side. The total velocity can be found using Pythagorean theorem:\(\sqrt{{(39000)^2 + (22000)^2}}\)=44000 kg km/h. The direction can be found using trigonometric function tangent inverse:\(\text{tan}^{-1}(\frac{22000}{39000})\)=29.5 degrees north of east. Hence, the total velocity and direction after collision are 44 km/h, 29.5 degrees north of east.
04

Validation of Part (a)

The answer from part (b) confirms the initial observation from part (a) that the unit of cars wouldn't move southeast, since after collision, the direction is proven to be north of east, not south. Therefore, the results from part (b) are consistent with the answer to part (a).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Collision Analysis
When two objects collide, such as cars at an intersection, understanding the collision well is crucial. This involves examining their movement directions and predicting the resultant motion after impact. In our exercise, one car travels north while the other heads east. After collision, they don't travel independently but as a single unit.
This combination moves in a direction resultant from their initial momenta. Given the movement directions (north and east), they will likely travel northeast post-collision.

To determine any impact on the southeast corner (as claimed by the property owner), we analyze the motion based on momentum principles. Since the combined motion veers between north and east, it suggests the southeast location remains unaffected. As a result, the property owner's claim lacks merit.
Momentum Calculation
Momentum is a key concept in understanding collisions. It's essentially the product of an object's mass and velocity. The conservation of momentum principle states that in an isolated system, the total momentum before collision equals the total momentum after.
  • Eastward Car: Mass = 1300 kg; Velocity = 30 km/h. Momentum = 1300 kg * 30 km/h = 39000 kg km/h.
  • Northward Car: Mass = 1100 kg; Velocity = 20 km/h. Momentum = 1100 kg * 20 km/h = 22000 kg km/h.

These calculations help us determine the combined movement. By summing the individual momenta components鈥攅ast and north鈥攚e can predict the unit's path after collision. In the exercise, the calculated total momentum gives us both speed and direction of the post-collision path.
Vector Resolution
To find the exact path of the cars post-collision, we resolve the momentum using vectors. Each car contributes a vector, with the eastward car providing the x-component and the northward car the y-component.

The total velocity vector can be calculated using the Pythagorean theorem: \[ \text{Total velocity} = \sqrt{(39000)^2 + (22000)^2} = 44000 \text{ kg km/h} \]
This result shows the magnitude of their combined velocity.
The direction is found using the tangent function:\[ \theta = \tan^{-1} \left( \frac{22000}{39000} \right) = 29.5^\circ \text{ north of east} \]
This vector approach confirms they move northeast, supporting the analysis that the southeast corner remains untouched.

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