Physics 9702/52 — October/November 2020

Background Concept

The suggested relationship links the trolley’s speed $v$ after moving a distance $d$ to the mass $m$ placed on the trolley. If the system starts from rest and the acceleration is approximately constant for the short travel, then kinematics gives

$$v^2 = u^2 + 2ad \quad (u=0) \Rightarrow v^2 = 2ad.$$

Rearranging gives

$$\frac{2d}{v^2} = \frac{1}{a}.$$

So a quantity like $\dfrac{2d}{v^2}$ behaves like the reciprocal of acceleration. In this experiment, the acceleration depends on the total inertia being accelerated (includes $m$ plus other effective masses) and the net driving force (weight of $P$ minus frictional effects, etc.), which is why the given model has constants $Q$ and $R$.

For finding constants experimentally, we aim to transform the equation into the straight-line form

$$y = Sx + C,$$

so that the gradient and intercept can be used to calculate unknowns.

Understanding the Question

You must design a practical procedure that:

• varies $m$ (mass on the trolley),
• measures the speed $v$ of the trolley when it has moved a fixed distance $d$ from its start,
• keeps other variables (especially the hanging mass $P$ and the distance $d$) constant,
• uses the results to test whether the given equation is correct, and
• extracts numerical values of the constants $Q$ and $R$.

Because the question asks for speed at a particular position ($d$ from the start), an instantaneous speed measurement at that point is ideal. A light gate + interrupt card provides that cleanly.

Approach

1. Choose a reliable way to measure $v$ at a known position: place a light gate at distance $d$ and attach an interrupt card of known length $L$ to the trolley.
2. For each chosen $m$, release the trolley from rest at the same start line and record the time for the card to pass the light gate. Compute $v=L/t$.
3. Repeat to reduce random error and obtain mean $v$ for each $m$.
4. Compute $y = 2d/v^2$ for each $m$ and plot $y$ against $m$.
5. Compare the graph with the linearised form of the model to obtain $Q$ and $R$ from the gradient and intercept.

Step-by-Step Reasoning

1) Variables

• Independent variable: $m$ (add/remove slotted masses on the trolley).
• Dependent variable: $v$ when the trolley reaches the position $d$.
• Controlled variables (examples that matter for the model):
  • $P$ (keep the hanging mass the same for all runs).
  • $d$ (fixed light gate position relative to the start line).
  • The surface/track and trolley (same wooden surface, same trolley wheels/condition).
  • The string and pulley arrangement (same pulley, string taut, similar alignment).
  • Release conditions: always start from rest at the same point; do not push.

2) Measuring $v$ at distance $d$

Attach an interrupt card of measured length $L$ to the trolley. Place the light gate exactly at the point $d$ from the start line.

When the card passes through, the light gate measures the blocking time $t$, giving the trolley’s instantaneous speed at that point:

$$v = \frac{L}{t}.$$

This is better than using a stopwatch over $d$ because the stopwatch gives an average speed and has large reaction-time error.

3) Collecting sufficient data

Use at least 6 values of $m$ spanning a sensible range (large enough to see a trend but not so large that motion becomes slow/jerky). For each $m$, repeat the run at least 3 times and average $v$.

Measuring $d$ with a metre rule and keeping it fixed is important because $d$ is used directly in the plotted quantity $2d/v^2$.

4) Linearising and using the graph

Start from the given equation:

$$\frac{2d}{v^2} = \frac{m + R}{Pg - Q}.$$

Rearrange into straight-line form by splitting the numerator:

$$\frac{2d}{v^2} = \frac{1}{Pg-Q}m + \frac{R}{Pg-Q}.$$

So, if you define

• $y = \dfrac{2d}{v^2}$,
• $x = m$,

then

$$y = Sx + C$$

with

$$S = \frac{1}{Pg-Q}, \quad C = \frac{R}{Pg-Q}.$$

From the best-fit straight line on the $y$ vs $m$ graph:

• find gradient $S$ and intercept $C$,
• calculate

$$Q = Pg - \frac{1}{S}$$

and

$$R = \frac{C}{S}$$

(using $C = R\,S$ since $S=1/(Pg-Q)$).

A straight line (within scatter) supports the proposed relationship.

5) Safety

Main hazards are the falling mass and moving trolley:

• Clamp pulley securely so it cannot detach.
• Ensure the hanging mass cannot strike feet or bounce dangerously (use a tray/soft landing; keep clear).
• Keep fingers away from the string and trolley path.

Key Takeaways

• A good plan identifies variables clearly and explains how controls are maintained.
• Instantaneous speed at a point is best measured using a light gate and interrupt card.
• To determine constants, linearise the model and use a graph’s gradient/intercept.
• Repeats and averaging are essential for reducing random uncertainties.

Common Mistakes

• Measuring average speed over $d$ with a stopwatch and calling it $v$ at distance $d$.
• Changing more than one variable (e.g. altering $P$ while varying $m$).
• Not keeping the start position the same, so $d$ is inconsistent.
• Plotting the wrong graph (e.g. $v$ vs $m$) which does not directly yield $Q$ and $R$.
• Forgetting to state how $Q$ and $R$ are obtained from gradient and intercept.

Things to Be Careful About

• Ensure the light gate is positioned so the trolley has actually travelled distance $d$ from the same reference point each run.
• Keep the string taut at release; slack changes the initial motion.
• Keep the added mass securely on the trolley so it does not slide (which would change effective motion).
• Use a wide range of $m$ values and enough points to justify a straight-line conclusion.
• When calculating $Q$, use the measured value of $P$ (in $\text{kg}$) and the given/known $g$ consistently in SI units.