Question

Meteorite On October 9,1992, a $12=\mathrm{kg}$ meteorite struck a car in Peekskill, New York, creating a dent about $22\mathrm{cm}$ deep, as shown in Figure 6.16. If the initial speed of the meteorite was $550\mathrm{m}/\mathrm{s}$, what was the average force exerted on the meteorite by the car?

Short answer

The average force exerted by the car on the meteorite is approximately 8,250,000 N.

Step-by-step solution

Understanding the Problem

We need to find the average force exerted on a meteorite by a car given that its initial speed is 550 m/s, its mass is 12 kg, and it created a 22 cm deep dent. The average force can be found using the work-energy principle.

Calculate Initial Kinetic Energy

Use the formula for kinetic energy: $KE=\frac{1}{2}m{v}^2$, where $m$ is the mass and $v$ is the initial speed of the meteorite. Thus, $KE=\frac{1}{2}\times 12\times (550{)}^2$.

Convert Units for Depth

Convert the depth of the dent from centimeters to meters: 22 cm = 0.22 m.

Use Work-Energy Principle

The work done by the car on the meteorite (which is equal to the force times the displacement) is equal to the change in kinetic energy. The final kinetic energy is zero, so the work done is $K{E}_{initial}=F\times 0.22$.

Solve for Average Force

Rearrange the equation to solve for $F$: $F=\frac{K{E}_{initial}}{0.22}$. Substitute the initial kinetic energy calculated in Step 2 to find $F$.

Calculating Final Force Value

Plug the initial kinetic energy and depth into the formula: $F=\frac{0.5\times 12\times {550}^2}{0.22}$, which equals $F=\frac{1815000}{0.22}$. Calculating this gives the average force exerted by the car.

Key concepts

Kinetic Energy

Kinetic Energy is the energy that an object possesses due to its motion. It depends on two main factors: the mass of the object and its velocity. The formula to calculate kinetic energy ($$KE$$) is given by:

• $KE=\frac{1}{2}m{v}^2$

where:

• $m$ represents the mass of the object
• $v$ represents the velocity of the object

In our exercise, the meteorite has a mass of 12 kg and an initial velocity of 550 m/s. Plugging these values into the formula gives us:

• $KE=\frac{1}{2}\times 12\times (550{)}^2$

This calculation yields the initial kinetic energy before the meteorite collides with the car, showing how much energy needs to be transformed during the collision.

Force Calculation

Calculating force can be done using the Work-Energy Principle. This principle states that the work done on an object is equal to the change in its kinetic energy. In the case of our meteorite, the work done by the car to stop it can be described as:

• $Work=Force\times Displacement$

Once the meteorite hits the car, it stops, meaning its final kinetic energy is zero. The change in kinetic energy, therefore, equals the initial kinetic energy. Therefore:

• $K{E}_{initial}=F\times 0.22$

Here, $F$ is the average force exerted by the car, and 0.22 m is the displacement (depth of the dent). By rearranging, you solve for force:

• $F=\frac{K{E}_{initial}}{0.22}$

This formula relates the energy transformed during the collision to the average force, offering insight into how the interaction occurs.

Unit Conversion

Unit Conversion helps in ensuring that all measurements are consistent before performing calculations. In physics, it's crucial to convert units properly for accuracy.
In the exercise, the initial depth of the dent was given in centimeters. Since the standard unit for displacement in physics calculations is meters, we must convert:

• 22 cm = 0.22 m

Unit conversion is essential because it maintains consistency in formulae that use standard units, ensuring calculations in formulas like Force are accurate.
Always double-check units in problems to prevent miscalculations, especially when dealing with systems that use different measurement units, such as metric and imperial.