June 2024 Paper 2 Q19
19 In this question use \(g = 9.8\) m s−2
A toy shoots balls upwards with an initial velocity of 7 m s−1
The advertisement for this toy claims the balls can reach a maximum height of 2.5 metres from the ground.
(a) Suppose that the toy shoots the balls vertically upwards.
(i) Verify the claim in the advertisement. [2 marks]
(ii) State two modelling assumptions you have made in verifying this claim. [2 marks]
(b) In fact the toy shoots the balls anywhere between 0 and 11 degrees from the vertical.
The range of maximum heights, \(h\) metres, above the ground which can be reached by the balls may be expressed as
\[k \lt h \leqslant 2.5\]Find the value of \(k\) [4 marks]
| Scheme | Marks | AO |
|---|---|---|
| (i) Substitutes three of the four given values into \(v^2 = u^2 + 2as\) Condone inconsistent signs for the substituted values | M1 | 3.1b |
| Completes reasoned argument to obtain the correct fourth value and concludes the claim is correct Must have clearly stated three out of the following four \(v = 0\) \(u = 7\) \(a = -9.8\) or \(-g\) \(s = 2.5\) Allow a consistent swapping of signs AG | R1 | 2.1 |
| (2) | ||
(ii) States one valid assumption from:
| E1 | 3.5b |
States a second valid assumption from
| E1 | 3.5b |
| (2) |
Typical solution
(i)
\[v = 0,\ u = 7,\ a = -9.8\]Using \(v^2 = u^2 + 2as\) then:
\[0 = 49 - 2(9.8)h\]\[h = \frac{49}{2(9.8)} = 2.5\text{ m}\]The claim is correct
(ii)
- The ball experiences no air resistance
- Ball projected from ground level
| Scheme | Marks | AO |
|---|---|---|
| States or uses 7cos11 for the vertical component of velocity OE | B1 | 2.2a |
| Uses \(v^2 = u^2 + 2as\) with \(u\) = their vertical component of velocity, \(v\) = 0 and \(a = -9.8\) | M1 | 3.1b |
| Obtains \(0 = 49\cos^2 11 + 2(-9.8)k\) Accept any variable for \(k\) | A1 | 1.1b |
| Obtains AWRT 2.4 | A1 | 1.1b |
| (4) | ||
| (8 marks) |