Physics 5054/21 — October/November 2015

Walkthrough

The question asks for the average acceleration of the arrow. Acceleration is defined as the change in velocity divided by the time taken for that change. The arrow starts from rest, so the initial velocity $u = 0 \text{ m/s}$ and the final velocity $v = 95 \text{ m/s}$. The time interval is $t = 0.011 \text{ s}$. Substituting these values into the formula $a = \frac{v - u}{t}$ gives $a = \frac{95}{0.011} = 8636.36... \text{ m/s}^2$. Rounding to two significant figures (matching the precision of the given values like 95 and 0.011), we get $8.6 \times 10^3 \text{ m/s}^2$.

Key Takeaways

Average acceleration is calculated using $a = \frac{\Delta v}{t}$. Always ensure the change in velocity is final minus initial, and units are consistent.

Common Mistakes

• Forgetting that the arrow starts from rest and using an incorrect initial velocity.
• Not converting units (though here all are already in standard SI units: m/s and s).
• Rounding too early in the calculation, which can lead to a slightly different force in part (ii).

Things to Be Careful About

• The mark scheme accepts $8.6 \times 10^3 \text{ m/s}^2$ or $8600 \text{ m/s}^2$. Using scientific notation is safer for large numbers.
• Keep the unrounded value ($8636.36...$) in your calculator for use in part (ii) to avoid rounding errors.

Walkthrough

Newton's second law states that the resultant force acting on an object is equal to its mass multiplied by its acceleration ($F = ma$). The mass of the arrow is given as $m = 0.018 \text{ kg}$. Using the average acceleration calculated in part (i), $a = 8636 \text{ m/s}^2$ (using the unrounded value for accuracy), we calculate the average force:

Rounding to appropriate significant figures, the average force is $155 \text{ N}$. The mark scheme also accepts values between 150 N and 160 N, reflecting the acceptance of slightly rounded acceleration values from part (i).

Key Takeaways

Newton's second law ($F = ma$) links force, mass, and acceleration directly. When using a calculated value from a previous part, it is best practice to use the unrounded value to minimise compounding errors.

Common Mistakes

• Using the rounded acceleration value ($8600$) instead of the more precise value ($8636$), leading to a force of $154.8 \text{ N}$, which is still acceptable but shows the importance of keeping extra digits.
• Forgetting to include the unit Newtons (N) in the final answer.
• Confusing mass and weight; here, mass is given directly in kg, so no multiplication by $g$ is needed.

Things to Be Careful About

• The mass is given in kg ($0.018 \text{ kg}$), which is the correct SI unit, so no conversion is needed.
• The mark scheme explicitly allows a range (150 to 160 N) because students might carry forward different levels of rounding from part (i). Always state your working clearly so the examiner can follow your error carried forward (ecf) if applicable.