If both the weight and speed of the vehicle are doubled, how many times must the braking force be increased to stop the vehicle?

To determine how many times the braking force must be increased to stop a vehicle when both its weight and speed are doubled, we can refer to the basic principles of physics, particularly Newton's second law of motion and the relationship between force, mass, and acceleration.

When the weight of the vehicle is doubled, the mass (m) of the vehicle doubles. According to the equation for kinetic energy (KE = 1/2 mv²), if the speed (v) of the vehicle is also doubled, the kinetic energy increases significantly. Specifically, if the speed is doubled, the kinetic energy becomes four times greater because kinetic energy is proportional to the square of the speed. This is calculated as follows:

1. Original kinetic energy is ( KE_1 = \frac{1}{2} mv^2 ).

2. New kinetic energy after doubling speed is ( KE_2 = \frac{1}{2} m(2v)^2 = \frac{1}{2} m(4v^2) = 2mv^2 ).

Thus, by doubling both mass and speed:

• The doubled mass contributes to a straightforward doubling of the resistance that must be overcome to stop the vehicle.

• The quadrupled energy due to