Problem structure
(a) 2, numerical, algebraic manipulation and units.
(b) 3, numerical, calculus/area under non-linear graph.
(c) 4, numerical, equations of motion (SUVAT).
(d) 4, numerical, multi-stage problem solving and stopping distance definitions.
Plan your route before writing. Use equations, diagrams, units, and a clear final justification where needed.
A high-speed "smart-sled" is tested on a horizontal track for emergency braking systems. The sled, initially travelling at a constant velocity $u$, passes a sensor at $t = 0$. At this instant, the braking system is engaged.
The velocity $v$ of the sled as a function of time $t$ is recorded by an onboard computer. For the first 2.00 seconds of braking, the velocity follows the relationship:
$$v(t) = u - kt^2$$
where $k$ is a constant of the braking system. After $t = 2.00$ s, the sled switches to a constant deceleration $a_c$ until it comes to a complete rest.
Data recorded:
- Initial velocity $u = 45.0\text{ m s}^{-1}$
- Velocity at $t = 2.00\text{ s}$ is $v_2 = 33.0\text{ m s}^{-1}$
- Total stopping distance from $t = 0$ is $D = 110\text{ m}$
(a)
2 marks
Determine the value and units of the constant $k$.
(b)
3 marks
Calculate the distance travelled by the sled during the first 2.00 seconds of braking.
(c)
4 marks
Determine the magnitude of the constant deceleration $a_c$ applied after $t = 2.00$ s.
(d)
4 marks
A safety engineer suggests that if the initial velocity $u$ were increased by 10%, the "thinking distance" (the distance travelled at constant $u$ before the brakes engage) would become significant. If the total stopping distance must not exceed 150 m, calculate the maximum allowable reaction time $t_r$ for this higher speed, assuming the braking profile ($k$ and $a_c$) remains identical.
