Question

1. an airplane accelerates down a runway at 3.20 m/s² for 32.8 s until it finally lifts off the ground. determine the distance traveled before takeoff.
2. a car starts from rest and accelerates uniformly over a time of 5.21 seconds for a distance of 110 m. determine the acceleration of the car.

Explanation:

Step1: Identify the relevant kinematic equation

For the first problem, we use the equation $x = v_0t+\frac{1}{2}at^{2}$, where $v_0 = 0$ (starts from rest), $a = 3.20\ m/s^{2}$ and $t = 32.8\ s$.
For the second problem, we use the same equation $x = v_0t+\frac{1}{2}at^{2}$, with $v_0 = 0$ (starts from rest), $x = 110\ m$ and $t = 5.21\ s$, and we solve for $a$.

Step2: Solve the first problem

Since $v_0 = 0$, the equation simplifies to $x=\frac{1}{2}at^{2}$. Substitute $a = 3.20\ m/s^{2}$ and $t = 32.8\ s$ into the equation:
$x=\frac{1}{2}\times3.20\times(32.8)^{2}$
$x = 1.6\times1075.84$
$x = 1721.344\ m$

Step3: Solve the second problem

Since $v_0 = 0$, the equation $x=\frac{1}{2}at^{2}$. We can re - arrange it to solve for $a$: $a=\frac{2x}{t^{2}}$.
Substitute $x = 110\ m$ and $t = 5.21\ s$ into the equation:
$a=\frac{2\times110}{(5.21)^{2}}$
$a=\frac{220}{27.1441}$
$a\approx8.105\ m/s^{2}$

Answer:

1. The distance traveled by the airplane before take - off is $1721.344\ m$.
2. The acceleration of the car is approximately $8.105\ m/s^{2}$.