Newton's Second Law — Question 1

Question 1

A net force determines acceleration, not velocity

Use $\sum\vec F=m\vec a$ for constant mass in an inertial frame. A 2 kg body has forces $(8,0)$ N and $(-2,4)$ N. Its initial velocity is $(-1,2)$ m/s. The forces remain constant for 3 s.

1. Find net force and acceleration.

2. Find velocity at 3 s.

3. Find displacement and explain why it is not generally parallel to the force.

Question 1 - Solution

A net force determines acceleration, not velocity

(a) Sum before dividing.

$$\vec F=(8,0)+(-2,4)=(6,4)\,\mathrm N,\quad\boxed{\vec a=\vec F/2=(3,2)\,\mathrm{m/s^2}}$$ $\vec F_{\mathrm{net}}=(6,4)$ N, so $\vec a=(3,2)$ m/s$^2$.

(b) Add the velocity change.

$$\vec v(3)=\vec v_0+\vec a(3)=(-1,2)+(9,6)=(8,8)\,\mathrm{m/s}$$ $\vec v(3)=(-1,2)+(3,2)3=(8,8)$ m/s.

(c) Include the initial-motion term.

$$\Delta\vec r=\vec v_0(3)+\tfrac12\vec a(3)^2=(-3,6)+(13.5,9)=(10.5,15)\,\mathrm m$$ $\Delta\vec r=\vec v_0t+\tfrac12\vec at^2=(-3,6)+(13.5,9)=(10.5,15)$ m. The acceleration is parallel to net force; velocity and displacement also depend on initial data and need not share that direction.