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Chapter 7: Problem 59

A small car of mass 1000 . kg traveling at a speed of \(33.0 \mathrm{~m} / \mathrm{s}\) collides head on with a large car of mass \(3000 \mathrm{~kg}\) traveling in the opposite direction at a speed of \(30.0 \mathrm{~m} / \mathrm{s}\). The two cars stick together. The duration of the collision is \(100 . \mathrm{ms}\). What acceleration (in \(g\) ) do the occupants of the small car experience? What acceleration (in \(g\) ) do the occupants of the large car experience?

Short Answer

Expert verified
Question: Determine the acceleration experienced by the occupants of both cars in terms of g, after following the given step-by-step solution.

Step by step solution

01

Determine initial momentum

The initial momentum of both cars have to be calculated. Momentum is the product of mass and velocity. For the small car, its initial momentum: \(p_{initial_{small}} = m_{small} * v_{initial_{small}}\) For the large car, its initial momentum: \(p_{initial_{large}} = m_{large} * -v_{initial_{large}}\), since it travels in the opposite direction.
02

Calculate the total initial momentum

Sum up the initial momentum values of both cars. \(p_{initial_{total}} = p_{initial_{small}} + p_{initial_{large}}\)
03

Determine the final momentum

Since they stick together, they move as a single object with a combined mass. Calculate the final momentum of the two-car system. \(p_{final_{total}} = (m_{small} + m_{large}) * v_{final}\)
04

Apply the conservation of momentum

According to the conservation of momentum principle, the total initial momentum is equal to the total final momentum. \(p_{initial_{total}} = p_{final_{total}}\)
05

Calculate the final velocity

Solve the equation from Step 4 to get the final velocity of the two-car system. \(v_{final} = \dfrac{p_{initial_{total}}}{m_{small} + m_{large}}\)
06

Calculate acceleration of each car

Using the final velocity and the duration of the collision, calculate the acceleration for each car. For the small car: \(a_{small} = \dfrac{v_{final} - v_{initial_{small}}}{t}\) For the large car: \(a_{large} = \dfrac{v_{final} - (-v_{initial_{large}})}{t}\)
07

Convert acceleration to g

To convert the acceleration values to g, divide each by the gravitational acceleration (9.81 m/s²). For the small car: \(g_{small} = \dfrac{a_{small}}{9.81}\) For the large car: \(g_{large} = \dfrac{a_{large}}{9.81}\) By following the steps, the acceleration experienced by the occupants of both cars in terms of g will be determined.

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