Newton's First Law — Question 3

Question 3

Straightness and uniform speed are separate tests

Three planar motions are observed in an inertial frame: A moves east at constant 4 m/s; B moves east with increasing speed; C travels around a circle at constant 4 m/s.

1. Which motion is compatible with zero net force throughout?

2. For B and C, identify what changes in the velocity vector.

3. Does an unchanged speed alone establish balanced forces?

Question 3 - Solution

Straightness and uniform speed are separate tests

(a) Test constant vector velocity.

$$\vec a_A=\frac{d(4\hat i)}{dt}=\vec0,\quad\boxed{\vec F_{{\mathrm{net}},A}=\vec0}$$ Only A has constant velocity and is compatible with zero resultant force throughout. B and C each have acceleration.

(b) Separate magnitude and direction.

$$\vec a=\dot v\,\hat t+v\frac{d\hat t}{dt}:\quad\dot v_B>0,\quad |\vec a_C|=\frac{4^2}{r}>0$$ B changes the magnitude of its eastward velocity. C keeps the magnitude but turns the direction continuously, so its velocity changes as well.

(c) Use both vector attributes.

$$\frac{d|\vec v|}{dt}=0\ \not\Rightarrow\ \frac{d\vec v}{dt}=\vec0$$ No. Constant speed does not imply constant velocity. The first-law condition concerns a straight, constant-speed trajectory, including rest as the zero-velocity case, in an inertial frame.