Newton's Second Law — Question 9

Question 9

Scaling mass and force together preserves acceleration

Use $\sum\vec F=m\vec a$ for constant mass in an inertial frame. Run A uses mass 2 kg and net force $(4,6)$ N. Run B uses mass 6 kg and force $(12,18)$ N. Both begin with the same position and velocity.

1. Compare accelerations.

2. Compare their trajectories while those forces persist.

3. Would doubling only mass preserve the original trajectory?

Question 9 - Solution

Scaling mass and force together preserves acceleration

(a) Divide each force by its own mass.

$$\boxed{\vec a_A=(4,6)/2=\vec a_B=(12,18)/6=(2,3)\,\mathrm{m/s^2}}$$ Both accelerations are $(2,3)$ m/s$^2$.

(b) Use identical evolution and initial data.

$$\vec r_B-\vec r_A=\Delta\vec r_0+\Delta\vec v_0t+\tfrac12\Delta\vec at^2=\vec0$$ Their velocity and position histories are identical over the specified interval. Equality of acceleration alone would not suffice if initial data differed.

(c) Change one parameter at a time.

$$\vec a'=\frac{\vec F}{2m}=\tfrac12\vec a,\quad\vec r'-\vec r=-\tfrac14\vec at^2$$ Doubling mass alone halves acceleration for the same net force. With the same initial data, displacement contributions from acceleration are halved; the original trajectory is generally not preserved.