Newtonian Mechanics — Question 8

Question 8

Initial conditions complete a motion prediction

A model stipulates constant acceleration $a=2$ m/s$^2$. Run A starts at $x=0$ with $v=1$ m/s. Run B starts at $x=5$ m with $v=-1$ m/s. Use the same time origin.

1. Write each position history.

2. Find their positions and velocities at 2 s.

3. Explain why one acceleration law does not predict a unique trajectory without initial data.

Question 8 - Solution

Initial conditions complete a motion prediction

(a) Integrate with the correct constants.

$$x=x_0+v_0t+\tfrac12at^2:\quad x_A=t+t^2,\quad x_B=5-t+t^2$$ $x_A=t+t^2$ m and $x_B=5-t+t^2$ m for numerical seconds. Velocities are $1+2t$ and $-1+2t$ m/s.

(b) Evaluate the distinct states.

$$\boxed{(x_A,v_A)_{t=2}=(6,5)},\quad\boxed{(x_B,v_B)_{t=2}=(7,3)}\quad\text{(SI)}$$ At 2 s, $(x_A,v_A)=(6\ \mathrm m,5\ \mathrm{m/s})$ and $(x_B,v_B)=(7\ \mathrm m,3\ \mathrm{m/s})$.

(c) Retain integration constants.

$$v(t)=v_0+\int_0^t a\,du,\qquad x(t)=x_0+\int_0^t v(u)\,du$$ Acceleration fixes how velocity changes. Initial velocity sets the starting rate, and initial position sets the spatial offset. Both constants are needed even when the dynamical rule itself is fixed.