Question

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

Short answer

The net force required is approximately -3,689,238 N.

Step-by-step solution

Convert Units of Speed and Distance

First, convert the initial speed from kilometers per hour to meters per second: \(45.0 \text{ km/h} = \frac{45.0 \times 1000 \text{ m}}{3600 \text{ s}} = 12.5 \text{ m/s}\)Then, convert the stopping distance from centimeters to meters:\(1.8 \text{ cm} = 0.018 \text{ m}\)

Determine the Required Acceleration

Use the formula for acceleration based on constant stopping distance, given by the equation:\(v_f^2 = v_i^2 + 2a x\)where **\(v_f\)** is the final velocity (0 m/s), **\(v_i\)** is the initial velocity (12.5 m/s), **\(a\)** is the acceleration (deceleration in this case), and **\(x\)** is the stopping distance (0.018 m). Since the car stops, \(v_f = 0\).We can rearrange to solve for **\(a\):**\[a = \frac{v_f^2 - v_i^2}{2x} = \frac{0 - 12.5^2}{2 \times 0.018} = \frac{-156.25}{0.036} = -4340.28 \text{ m/s}^2\]

Calculate the Net Force

Using Newton's second law, the net force can be calculated with the equation:\(F = m \cdot a\)Substitute the given mass (850 kg) and the calculated acceleration (-4340.28 m/s²) into the equation:\[F = 850 \text{ kg} \times (-4340.28 \text{ m/s}^2) = -3,689,238 \text{ N}\]The negative sign indicates that the force is in the opposite direction of the motion.

Key concepts

Newton's second law

Newton's second law of motion is a fundamental principle that describes how the velocity of an object changes when it is subjected to an external force. This law is typically expressed in the equation:
\( F = m \cdot a \)
This means that the net force \( F \) acting on an object is equal to the mass \( m \) of the object multiplied by its acceleration \( a \). This relationship helps us understand how forces influence motion.
For example, when calculating the force needed to stop a car, like in the exercise, this law allows us to relate the car's mass and its deceleration to the force applied. It’s important to recognize that the force acting on an object not only depends on how much the object weighs but also how quickly it needs to stop or accelerate.

Acceleration calculation

Acceleration is defined as the rate of change of velocity of an object. It measures how quickly speed increases or decreases, and is given in meters per second squared (\( m/s^2 \)).
In problems involving motion, acceleration can be calculated using the formula:
\[ a = \frac{v_f^2 - v_i^2}{2x} \]
Here, \( v_f \) is the final velocity, \( v_i \) is the initial velocity, and \( x \) is the distance over which the change occurs.
To find the needed deceleration for a car to stop "on a dime," we substitute \( v_f = 0 \) (since the car stops), \( v_i = 12.5 \text{ m/s} \), and \( x = 0.018 \text{ m} \). The resulting acceleration is negative, indicating deceleration, or slowing down.

Unit conversion

Unit conversion is crucial in physics to ensure that calculations are accurate and meaningful. In this exercise, converting units involves transforming the initial velocity from kilometers per hour to meters per second, and the stopping distance from centimeters to meters.
To convert kilometers per hour to meters per second, use the relation:

• 1 kilometer = 1000 meters
• 1 hour = 3600 seconds

Hence, \[ 45.0 \text{ km/h} = \frac{45.0 \times 1000}{3600} = 12.5 \text{ m/s} \]
Similarly, for converting centimeters to meters:

• 1 cm = 0.01 m

Thus, \[ 1.8 \text{ cm} = 0.018 \text{ m} \]
These conversions ensure that all quantities are in compatible units for calculations.

Deceleration

Deceleration is simply negative acceleration, meaning an object is slowing down rather than speeding up. It is measured in the same units, \( m/s^2 \).
In the context of the exercise, calculating deceleration involves understanding the change in velocity over the stopping distance. Here, the initial speed of the car needs to be reduced to zero over a very short distance, which requires a very large negative acceleration.
This large negative value indicates the car needs to slow down rapidly, highlighting the difficulty of stopping quickly over small distances like the diameter of a dime.

Stopping distance

Stopping distance is the distance required for a vehicle or object to come to a complete stop from a specific initial velocity. It is a crucial factor in safety assessments, such as determining safe following distances while driving.
In physics problems, stopping distance is often part of the equation where variables like initial speed and deceleration are known. The stopping distance can indicate how quickly an object can be brought to a halt. This relationship is captured in the formula used to calculate acceleration:
\[ 0 = v_i^2 + 2a x \]
For an automobile to "stop on a dime," very precise and quick deceleration is necessary. In reality, very short stopping distances require an immense force, which has practical limitations.