Airplane Landing Deceleration and Position

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Kinematics Beginner linear motion

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.

Find the distance of the airplane from its landing point 12 seconds after landing.

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.