Acceleration due to Gravity

• Gravity is a vector quantity that always acts vertically downwards toward the center of the Earth. This directional consistency is crucial when setting up coordinate systems for vertical motion problems.

• The motion is modeled using constant acceleration, which allows for the use of the five suvat equations. Because $a$ is fixed as $\pm g$, the complexity of the problem is reduced to identifying initial conditions and the desired final state.

• At the maximum height of a vertical trajectory, the instantaneous velocity of the object is exactly $0 \text{ m s}^{-1}$. This principle provides a hidden piece of data ($v=0$) that is essential for solving problems involving peak height or total time of flight.

• Establish a Sign Convention: Before performing any calculations, you must decide which direction (up or down) is positive. If upwards is positive, then $a = -9.8 \text{ m s}^{-2}$; if downwards is positive, then $a = +9.8 \text{ m s}^{-2}$.

• Identify Known Variables: List the known suvat values ($s, u, v, a, t$) based on the problem description. Common indicators include 'dropped' ($u=0$), 'returns to ground' ($s=0$ if starting from ground), and 'reaches peak' ($v=0$).

• Select the Equation: Choose the suvat equation that links your three known variables to the one unknown variable you are trying to find. For example, use $v^2 = u^2 + 2as$ to find displacement when time is not provided.

• Solve and Interpret: Perform the algebraic manipulation to isolate the unknown. Ensure the final answer is interpreted in the context of the chosen sign convention (e.g., a negative displacement means the object is below the starting point).

Displacement vs. Distance

• Displacement ($s$) is a vector representing the change in position relative to the starting point. If an object is thrown up and caught at the same height, its total displacement is zero, regardless of how high it traveled.
• Distance is a scalar representing the total path length covered. In the same throw-and-catch scenario, the distance would be twice the maximum height reached.

• Significant Figures: When using $g = 9.8$ (which is 2 significant figures), your final numerical answer should generally be rounded to 2 significant figures. Providing excessive precision (e.g., 4 or 5 decimal places) can result in a loss of marks for over-accuracy.

• Working in Terms of $g$: In multi-step problems, it is often more accurate and elegant to leave $g$ as a symbol until the final step. This prevents rounding errors from accumulating and sometimes allows $g$ to cancel out entirely.

• Sanity Checks: Always verify if your answer makes physical sense. For instance, the time of flight for a ball thrown by a human should typically be in seconds, not milliseconds or hours, and the impact speed should be greater than the speed at the peak.

• The 'Zero Velocity at Ground' Error: A very common mistake is assuming the final velocity ($v$) is zero when an object hits the ground. In mechanics, $v$ refers to the velocity the instant before impact; the impact itself is a separate event that involves external forces.

• Sign Inconsistency: Students often mix sign conventions, such as using a positive initial velocity ($u$) for an upward throw but also a positive value for $g$. This implies gravity is pushing the object upward, leading to incorrect results.

• Confusing $g$ with Force: Remember that $g$ is an acceleration ($9.8 \text{ m s}^{-2}$), not a force. While gravity causes the acceleration, the value $9.8$ should be used in the acceleration slot ($a$) of the suvat equations, not as a force in $F=ma$ unless multiplied by mass.