Problem 47 - College Physics

Question

An advertisement claims that a particular automobile can "stop on a dime." What net force would actually be necessary to stop a 850 -kg automobile traveling initially at 45.0 \(\mathrm{km} / \mathrm{h}\) in a distance equal to the diameter of a dime, which is 1.8 \(\mathrm{cm} ?\)

Step-by-Step Solution

Answer

Approximately 3,684,896 N of force is needed.

Step 1: Convert Units

Convert the speed of the car from kilometers per hour to meters per second. Since 1 km/h = 0.27778 m/s, multiply the speed by this conversion factor: \[45.0 \text{ km/h} \times 0.27778 \approx 12.5 \text{ m/s}.\] Also convert the diameter of a dime from centimeters to meters: \[1.8 \text{ cm} = 0.018 \text{ m}.\]

Step 2: Calculate Initial Velocity in m/s

The initial velocity \( v_i \) is already calculated as \( 12.5 \text{ m/s} \) from the previous step.

Step 3: Use the Work-Energy Principle

The work done by the net force to stop the car is equal to the change in kinetic energy. The initial kinetic energy \( KE_i \) is \( \frac{1}{2} m v_i^2 \), where \( m = 850 \text{ kg} \) and \( v_i = 12.5 \text{ m/s} \). Calculate \( KE_i \): \[KE_i = \frac{1}{2} \times 850 \times (12.5)^2.\]

Step 4: Calculate Initial Kinetic Energy

Solve the calculation for initial kinetic energy: \[KE_i = \frac{1}{2} \times 850 \times 156.25 = 66328.125 \text{ J} .\]

Step 5: Calculate Net Force Using Work Formula

The work done is equal to the force times the stopping distance (dime's diameter). So, \[F \cdot d = KE_i.\] Rearrange to find \( F \): \[F = \frac{KE_i}{d} = \frac{66328.125}{0.018}.\]

Step 6: Solve for Net Force

Calculate \( F \): \[ F \approx \frac{66328.125}{0.018} \approx 3684895.83 \text{ N}.\]

Step 7: Interpret the Result

The net force required to stop the car in the given distance is extremely large, suggesting it’s practically impossible with everyday forces.

Key Concepts

Kinetic energy

Kinetic energy is the energy an object possesses due to its motion. Understanding it is crucial to solving many physics problems. In simpler terms, think of kinetic energy as the effort needed to get something moving and keep it moving. When you push a swing, you're adding energy to it, giving it kinetic energy.

For a moving object, the formula to calculate kinetic energy (\(KE\)) is:

• \[ KE = \frac{1}{2} mv^2 \]

Here:

• \(m\) is the mass of the object (in kilograms).
• \(v\) is the velocity of the object (in meters per second).

In our problem, an 850 kg car moving at 12.5 m/s has its kinetic energy calculated as \(66328.125\) joules. This is the amount of energy it has just because it's moving!

The concept is vital because it shows how much work will be needed to change the object’s motion. To stop the car, its kinetic energy must be reduced to zero.

Work-energy principle

The work-energy principle connects the concepts of work and energy. It states that the work done on an object is equal to the change in its kinetic energy. In simpler terms, if you want to change an object's motion, you have to do work on it.

Work is calculated using the formula:

• \[ W = F \cdot d = \Delta KE \]

Here:

• \(W\) is the work done on the object.
• \(F\) is the force applied (in newtons).
• \(d\) is the distance over which the force is applied (in meters).
• \(\Delta KE\) is the change in kinetic energy.

In the given exercise, to stop the car in such a short distance — the size of a dime (0.018 m) — requires doing an immense amount of work equivalent to the car's initial kinetic energy (66328.125 J). The work-energy principle helps us figure out the force needed once the stopping distance is known.

By rearranging the formula to solve for force, you see how much force it actually takes to stop that movement.

Unit conversion

Unit conversion is a crucial skill in physics, especially when dealing with calculations that require consistent units. In our example, we need to ensure all distances, velocities, and masses are in the correct units for the formulas to work seamlessly.

The speed of the car is given in kilometers per hour (km/h), but for calculations, we often need it in meters per second (m/s). It's converted using:

• \[ v = 45.0 \, \text{km/h} \times 0.27778 \approx 12.5 \, \text{m/s} \]

Similarly, the diameter of a dime is converted from centimeters to meters:

• \[ 1.8 \, \text{cm} = 0.018 \, \text{m} \]

This skill ensures that each part of the equation uses compatible units, which is essential for correct results. Without proper unit conversion, even the right formulas will yield the wrong answers.

Force calculation

Force calculation is about determining how much push or pull is required to achieve a certain effect on an object, like stopping it or accelerating it. The force needed can often be immense, especially over short distances.

In this problem, we're calculating the net force required to stop a car. Using the work-energy principle:

• \[ F \cdot d = KE_i \]

Rearranged to find force:

• \[ F = \frac{KE_i}{d} \]
• Where \(KE_i\) is the initial kinetic energy and \(d\) is the stopping distance (0.018 m).

By substituting the given values, we find a staggering force of approximately 3684895.83 N is required to stop the car in the diameter of a dime. Such a force is impractical in everyday situations, illustrating both the power of physics calculations and the dramatic results possible when big energies and small distances are involved.

Mastering force calculations helps predict whether a given action is feasible within realistic scenarios.