Question

A car, of mass \(m,\) traveling at a speed \(v_{1}\) can brake to a stop within a distance \(d\). If the car speeds up by a factor of \(2, v_{2}=2 v_{1},\) by what factor is its stopping distance increased, assuming that the braking force \(F\) is approximately independent of the car's speed?

Short answer

Answer: The stopping distance increases by a factor of 4 when the car's speed is doubled.

Step-by-step solution

Calculate the initial kinetic energy of the car

First, we need to write down the formula for kinetic energy, which is given by \[K = \frac{1}{2}mv^2\] Let's calculate the initial kinetic energy of the car at both \(v_1\) and \(v_2\). When the car is traveling at \(v_1\), the kinetic energy is: \[K_1 = \frac{1}{2}mv_1^2\] When the car speeds up to \(v_2\), the kinetic energy becomes: \[K_2 = \frac{1}{2}m(2v_1)^2 = \frac{1}{2}m(4v_1^2) = 4K_1\]

Calculate the work done by the braking force

The work-energy principle states that the work done on an object is equal to the change in its kinetic energy. The car's kinetic energy is reduced to zero when it comes to a stop. Thus, the work done by the braking force is equal to the initial kinetic energy of the car. So, for the initial speed \(v_1\) and stopping distance \(d\), we have: \[W_1 = -F \cdot d = K_1\] For the new speed \(v_2\) and stopping distance \(d_2\), we have: \[W_2 = -F \cdot d_2 = K_2\]

Relating the stopping distances

We can now relate the stopping distances for the two different speeds using the equations for work. \[W_2 = 4W_1\] Since braking force \(F\) is approximately independent of the car's speed, we can write the above equation as: \[-F \cdot d_2 = 4(-F \cdot d)\] The braking force \(F\) can now be cancelled from both sides of the equation, leaving: \[d_2 = 4d\]

Conclusion

The stopping distance increases by a factor of 4 when the car's speed is doubled, i.e., \(v_2 = 2v_1\).

Key concepts

Kinetic Energy

Kinetic energy is a term that describes the energy an object possesses due to its motion. It is quantified by the equation
\( K = \frac{1}{2}mv^2 \),
where \( K \) is the kinetic energy, \( m \) is the mass of the object, and \( v \) is its velocity. In our exercise, this concept plays a crucial role in determining how far a car will travel before stopping.

Revisiting the Problem

In our scenario, we had a car with mass \(m\) moving at an initial speed \(v_1\). The kinetic energy at this speed was
\( K_1 = \frac{1}{2}mv_1^2 \).
But when the car's speed was doubled, \(v_2 = 2v_1\), its kinetic energy quadrupled, due to the squared term in the kinetic energy formula:
\( K_2 = 4K_1 \).
So, kinetic energy is not just about speed, but how changes in speed can lead to exponential increases in the energy involved, which impacts stopping distances significantly.

Work-Energy Principle

The work-energy principle is a powerful tool in physics that relates the work done on an object to its change in kinetic energy. The general statement of this principle is that the work done by all forces acting on an object equals the change in its kinetic energy.

Applying the Principle

In the context of our problem, this principle helps us understand how the car comes to a stop. The work done by the braking force, \(W\), is equal to the negative change in kinetic energy of the car as it slows down to a halt. Mathematically, this is expressed as
\(W = - \Delta K\).
This equation tells us that the work done by the braking force over the distance \(d\), which is equal to the initial kinetic energy of the car, \(K_1\), will bring the vehicle to a stop. Similarly, for the higher speed, the work \(W_2\) done over the stopping distance \(d_2\) is equal to the higher kinetic energy \(K_2\), determining the new stopping distance.

Braking Force

Braking force is the resistive force applied by a braking system to halt the motion of a vehicle. It's an essential factor in calculating the stopping distance for a car as it undergoes deceleration.

How Braking Force Factors In

While the braking force can be complex to calculate, due to factors such as friction and the condition of the brakes, in our problem, it is assumed to be constant regardless of the car's speed. This simplification allows us to focus on the relationship between kinetic energy and stopping distance without worrying about variabilities in the braking force.
Under this assumption, doubling the speed of the car quadruples the kinetic energy, but since the braking force remains constant, the car's stopping distance also quadruples to dissipate the additional kinetic energy. This direct relationship provides a digestible understanding of how changes in speed affect the work the braking force must do to stop a car.