Source: High school physics (Chinese)
Problem
An airplane lands with an initial velocity of 60 m/s and decelerates uniformly at a rate of 6 m/s².
The distance of the airplane from its landing point 12 seconds after landing is 300 m. Key formulas used: Time to stop: $t_{stop} = -\frac{v_0}{a}$ Position: $x = v_0 t + \frac{1}{2}at^2$
The motion of the airplane is described by the equations of kinematics for constant acceleration. The initial velocity is $v_0 = 60$ m/s and the acceleration is $a = -6$ m/s² (negative as it is a deceleration).
First, we determine the time required for the airplane to come to a complete stop ($v=0$). The velocity equation is:
$$v = v_0 + at$$Setting $v=0$ gives the stopping time, $t_{stop}$:
$$0 = v_0 + at_{stop}$$ $$t_{stop} = -\frac{v_0}{a}$$Substituting the values:
$$t_{stop} = -\frac{60 \text{ m/s}}{-6 \text{ m/s}^2} = 10 \text{ s}$$The question asks for the position at $t = 12$ s. Since this is after the airplane has come to a stop at $t_{stop} = 10$ s, the airplane's position will be its final stopping distance.
The position $x$ is given by the kinematic equation:
$$x = v_0 t + \frac{1}{2}at^2$$We evaluate this equation at the time of stopping, $t = t_{stop} = 10$ s, to find the stopping distance, $x_{stop}$.
$$x_{stop} = v_0 t_{stop} + \frac{1}{2}a t_{stop}^2$$Substituting the values:
$$x_{stop} = (60 \text{ m/s})(10 \text{ s}) + \frac{1}{2}(-6 \text{ m/s}^2)(10 \text{ s})^2$$ $$x_{stop} = 600 \text{ m} - 3(100) \text{ m} = 300 \text{ m}$$The position of the airplane at 12 seconds is this final stopping distance.