Problem structure
(a) 2, numerical, kinematics (projectile motion).
(b) 2, numerical, kinematics (projectile motion).
(c) 3, numerical, circular motion (centripetal force).
(d) 3, numerical, oscillations (SHM period).
(e) 2, numerical, oscillations (SHM acceleration).
Plan your route before writing. Use equations, diagrams, units, and a clear final justification where needed.
A small robotic probe of mass $m = 0.450\text{ kg}$ is designed to test the surface of a low-gravity asteroid. The probe is launched horizontally from the top of a vertical cliff of height $H = 12.0\text{ m}$ with an initial speed $u$.
(a)
2 marks
On this asteroid, the acceleration due to gravity is $g_{ast} = 1.60\text{ m s}^{-2}$. Calculate the time $t$ it takes for the probe to reach the ground, assuming no atmosphere.
(b)
2 marks
The probe must land at a horizontal distance (range) of $x = 18.0\text{ m}$ from the base of the cliff. Calculate the required horizontal launch speed $u$.
(c)
3 marks
Upon landing, the probe is immediately tethered to a fixed point on the ground and performs a horizontal circular sweep to scan the area. It moves in a circle of radius $r = 5.00\text{ m}$ at a constant speed. The maximum horizontal centripetal force the tether can withstand is $2.25\text{ N}$. Calculate the maximum angular velocity $\omega$ the probe can maintain without the tether snapping.
(d)
3 marks
The probe's internal sensor uses a small mass on a spring to detect vibrations. The sensor behaves as a simple harmonic oscillator with a period $T = 0.850\text{ s}$. If the mass of the sensor's internal block is $0.0250\text{ kg}$, calculate the stiffness (spring constant) $k$ of the spring.
(e)
2 marks
During a test, the sensor mass is displaced by $15.0\text{ mm}$ from its equilibrium position and released. Calculate the maximum acceleration $a_{max}$ of the sensor mass.
