Let’s solve the questions one by one.

(a) (i)

• Given:
  • Distance = $46 \, \text{cm} = 0.46 \, \text{m}$
  • Time = $6 \, \text{ns} = 6 \times 10^{-9} \, \text{s}$
• To Find: Uniform speed $v$
• Solution: $v = \frac{\text{Distance}}{\text{Time}}$ $v = \frac{0.46}{6 \times 10^{-9}} = 7.67 \times 10^7 \, \text{m/s}$

(a) (ii)

• Given:
  • The golf ball is released from rest ($u = 0$).
  • Acceleration due to gravity $g = 9.8 \, \text{m/s}^2$.
  • Time $t = 1 \, \text{s}$.
• To Find:
  1. Position $s$ after $1 \, \text{s}$.
  2. Velocity $v$ after $1 \, \text{s}$.
• Solution: Using equations of motion: $s = ut + \frac{1}{2} g t^2$ $s = 0 + \frac{1}{2} (9.8) (1^2) = 4.9 \, \text{m}$ $v = u + g t$ $v = 0 + 9.8 (1) = 9.8 \, \text{m/s}$

(b)

Given: $v = m + nt^2, \quad m = 10 \, \text{m/s}, \, n = 2 \, \text{m/s}^3$

(i) Change in velocity

• At $t_1 = 2 \, \text{s}$: $v_1 = 10 + 2(2^2) = 10 + 8 = 18 \, \text{m/s}$
• At $t_2 = 5 \, \text{s}$: $v_2 = 10 + 2(5^2) = 10 + 50 = 60 \, \text{m/s}$
• Change in velocity: $\Delta v = v_2 - v_1 = 60 - 18 = 42 \, \text{m/s}$

(ii) Average acceleration

• Formula: $\text{Average acceleration} = \frac{\Delta v}{\Delta t}$
• Solution: $a_{\text{avg}} = \frac{42}{5 - 2} = \frac{42}{3} = 14 \, \text{m/s}^2$

(iii) Instantaneous acceleration at $t = 2 \, \text{s}$

• Acceleration is the derivative of velocity: $a = \frac{d}{dt} (m + nt^2)$ $a = 0 + 2n t$
• At $t = 2 \, \text{s}$: $a = 2(2)(2) = 8 \, \text{m/s}^2$

(c)

• Given:
  • Initial velocity $u = 11 \, \text{m/s}$
  • Angle of projection $\theta = 20^\circ$
  • Acceleration due to gravity $g = 9.8 \, \text{m/s}^2$
• To Find: Horizontal range ($R$).
• Solution: Horizontal range is given by: $R = \frac{u^2 \sin 2\theta}{g}$ $\sin 2\theta = \sin(2 \times 20^\circ) = \sin 40^\circ = 0.6428$ $R = \frac{(11)^2 (0.6428)}{9.8}$ $R = \frac{121 (0.6428)}{9.8} = \frac{77.78}{9.8} = 7.94 \, \text{m}$

Would you like further details or clarification?

Related Questions

1. What factors affect the velocity of a golf ball in free fall?
2. How does the angle of projection impact the range of a projectile?
3. What is the significance of instantaneous acceleration in motion problems?
4. Can we find the time of flight for the long jumper? How?
5. How would the motion change if air resistance is considered?

Tip: Always break down problems into known formulas and steps to minimize confusion.