Kinetic Energy — Question 6

Question 6

A squared-speed plot separates mass from a sensor offset

A sensor reads $R=\tfrac12 m v^2+b$, where b is an unknown constant offset in joules. The three exact measurements below concern the same particle.

1. Find m and b.

2. Find the true kinetic energy at 4 m/s.

3. Predict R at rest and explain why forcing the graph through the origin would misestimate mass.

Question 6 - Solution

A squared-speed plot separates mass from a sensor offset

(a) Use differences to eliminate the offset.

The governing relation is $$\boxed{R=s v^2+b,\qquad s=\frac{\Delta R}{\Delta(v^2)}=\frac m2}$$ Eliminate the unknown offset by subtracting two readings: $$\frac m2=\frac{29-11}{16-4}=1.5\ \mathrm{kg}\Longrightarrow\boxed{m=3\ \mathrm{kg}}.$$ $$\boxed{b=11-\tfrac12(3)(4)=5\ \mathrm J},\qquad R(36)=\tfrac12(3)(36)+5=59\ \mathrm J.$$ The third point independently agrees with the inferred model.

(b) Remove the sensor offset.

At $v=4$ m/s, subtract the sensor offset: $$\boxed{K=\tfrac12(3)(4)^2=24\ \mathrm J},\qquad R=K+b=29\ \mathrm J.$$

(c) Evaluate zero speed.

The physical kinetic energy vanishes at rest, but the reading does not: $$K(0)=0,\qquad R(0)=b=5\ \mathrm J.$$ Forcing a zero intercept through the first point instead gives $$m_{\mathrm{false}}=\frac{2R}{v^2}=\frac{2(11)}4=5.5\ \mathrm{kg}\ne3\ \mathrm{kg}.$$ That fit assigns the offset to the mass.