According to Newton's First and Second Laws, if two vehicles of different masses arrive at an uncontrolled intersection at the same time and both apply the same braking force ($F$), which vehicle will come to a stop in a shorter distance?

• A. The vehicle with the smaller mass ($m$) will stop in a shorter distance because it will experience a greater acceleration ($a$) for the same force ($F$).
• B. Both vehicles will stop in the same distance because the braking force ($F$) is the same.
• C. Neither vehicle will stop; they will both continue moving at constant velocity.
• D. The vehicle with the larger mass ($m$) will stop in a shorter distance because it has more inertia.

Answer: (A)

Verified step by step guidance

1. Recall Newton's Second Law, which states that the acceleration \(a\) of an object is given by the net force \(F\) applied divided by its mass \(m\): \[ a = \frac{F}{m} \]
2. Since both vehicles apply the same braking force \(F\), the vehicle with the smaller mass \(m\) will experience a larger acceleration in magnitude (greater deceleration) because acceleration is inversely proportional to mass.
3. Use the kinematic equation relating initial velocity \(v_0\), acceleration \(a\), and stopping distance \(d\) when the final velocity \(v\) is zero: \[ v^2 = v_0^2 + 2 a d \] Rearranged for stopping distance \(d\): \[ d = -\frac{v_0^2}{2 a} \] Note that \(a\) is negative since it is a deceleration.
4. Since the vehicle with smaller mass has a larger magnitude of acceleration (greater deceleration), substituting into the equation shows it will have a smaller stopping distance \(d\) compared to the vehicle with larger mass.
5. Therefore, the vehicle with the smaller mass will come to a stop in a shorter distance when the same braking force is applied to both.