Hint
Choose and retain a positive direction; distinguish signed quantities from magnitudes. Write one equation in each particle’s direction of motion and eliminate tension.
Worked solution
(a)
For the heavier mass, \(80-T=8a\). For the lighter, \(T-60=6a\). Adding eliminates T and gives \(\displaystyle a=\)\(\displaystyle \frac{10}{7}\) \(\displaystyle \frac{m}{s^{2}}\).
Answer: \(\displaystyle a=\frac{10}{7}\) \(\displaystyle \frac{m}{s^{2}}\).
(b)
Substitute the common acceleration into a single-particle equation.
For the rising lighter particle, \(\displaystyle T=L(g+a)=\)\(\displaystyle \frac{480}{7}\) N.
Answer: \(\displaystyle \frac{480}{7}\) N
(c)
Use the common constant acceleration and elapsed time before any impact.
Starting from rest, v=at=\(\displaystyle \frac{40}{7}\) m/s.
Answer: \(\displaystyle \frac{40}{7}\) m/s
Maths4U indicative marking guidance
Maths4U indicative marking guidance, not an official mark scheme. Credit complete valid alternative methods and consequential reasoning. An unexplained result does not earn a method mark.
(a) · 3 marks
- 1 mark: Apply Newton’s second law to each particle with consistent opposite positive axes.
- 1 mark: Eliminate tension to determine the common acceleration.
- 1 mark: Give the complete requested result, using the required units, precision and boundary inclusion.
Alternative methods: Any complete mathematically valid alternative method with the same restrictions and equivalent exact result is acceptable.
Common error: Use one common tension only for the stated light string and smooth pulley.
(b) · 2 marks
- 1 mark: Substitute the common acceleration into a single-particle equation.
- 1 mark: Give the correct requested result, retaining signs, units and any domain restrictions.
Alternative methods: Any complete mathematically valid alternative method with the same restrictions and equivalent exact result is acceptable.
Common error: The connected particles move in opposite directions even though their acceleration magnitudes match.
(c) · 2 marks
- 1 mark: Use the common constant acceleration and elapsed time before any impact.
- 1 mark: Give the correct requested result, retaining signs, units and any domain restrictions.
Alternative methods: Any complete mathematically valid alternative method with the same restrictions and equivalent exact result is acceptable.
Common error: Use one common tension only for the stated light string and smooth pulley.
Total: 7 marks.