1. **State the problem:** We have data for speed $v$ (m/s) at different times $t$ (s) for a sensor dropped on a new planet. The equation relating speed and time is $$v = u + at$$ where $u$ is initial speed, $a$ is acceleration due to gravity, and $t$ is time. 2. **Identify variables:** $v$ is the dependent variable (speed changes with time), $t$ is the independent variable. 3. **Data points:** $(t,v)$ = (3,47), (5,81), (9,131), (12,194), (15,213), (16,268), (19,302). 4. **Find line of best fit:** We assume a linear relation $v = u + at$. 5. **Calculate slope $a$ (acceleration) and intercept $u$ (initial speed):** Calculate means: $$\bar{t} = \frac{3+5+9+12+15+16+19}{7} = \frac{79}{7} \approx 11.29$$ $$\bar{v} = \frac{47+81+131+194+213+268+302}{7} = \frac{1236}{7} \approx 176.57$$ Calculate slope: $$a = \frac{\sum (t_i - \bar{t})(v_i - \bar{v})}{\sum (t_i - \bar{t})^2}$$ Calculate numerator: $$(3-11.29)(47-176.57) + (5-11.29)(81-176.57) + (9-11.29)(131-176.57) + (12-11.29)(194-176.57) + (15-11.29)(213-176.57) + (16-11.29)(268-176.57) + (19-11.29)(302-176.57)$$ $$= (-8.29)(-129.57) + (-6.29)(-95.57) + (-2.29)(-45.57) + (0.71)(17.43) + (3.71)(36.43) + (4.71)(91.43) + (7.71)(125.43)$$ $$= 1074.5 + 601.1 + 104.3 + 12.4 + 135.2 + 430.5 + 966.9 = 3315.9$$ Calculate denominator: $$(3-11.29)^2 + (5-11.29)^2 + (9-11.29)^2 + (12-11.29)^2 + (15-11.29)^2 + (16-11.29)^2 + (19-11.29)^2$$ $$= 68.7 + 39.6 + 5.2 + 0.5 + 13.8 + 22.2 + 59.5 = 209.5$$ So slope: $$a = \frac{3315.9}{209.5} \approx 15.83$$ 6. **Calculate intercept $u$:** $$u = \bar{v} - a \bar{t} = 176.57 - 15.83 \times 11.29 = 176.57 - 178.7 = -2.13$$ 7. **Interpretation:** - Acceleration due to gravity $a \approx 15.83$ m/s$^2$. - Initial speed $u \approx -2.13$ m/s (small negative, likely measurement noise, effectively near zero). 8. **Equation of motion:** $$v = -2.13 + 15.83 t$$ 9. **Second sensor dropped at 50 m/s faster initial speed:** New initial speed $u_2 = u + 50 = -2.13 + 50 = 47.87$ m/s. Equation: $$v = 47.87 + 15.83 t$$ 10. **Same experiment on another planet with same initial speed:** If acceleration $a_2$ is unknown, equation is: $$v = -2.13 + a_2 t$$ --- **Summary:** - Acceleration due to gravity on the planet: $15.83$ m/s$^2$ - Initial speed of sensor: approximately $-2.13$ m/s - Equation of motion: $$v = -2.13 + 15.83 t$$ - Second sensor line: $$v = 47.87 + 15.83 t$$ - Another planet equation: $$v = -2.13 + a_2 t$$