Answer to Question #177176 in Classical Mechanics for Joseph Se

Question

The final answer should be in the format of "tolerance of ± in the third significant digit."

1.The drag racer P starts from rest at the start line S and then accelerates along the track. When it has traveled 87 m, its speed is 39 m/s and it is accelerating forward at 10.1 m/s². For that instant, determine the values of r¨ and θ¨ relative to axes fixed to an observer O in the grandstand G as shown.

28 meters from point O ( it is the base of the triangle)

 r¨ = ____ m/s²

 θ¨ = _____ rad/s²

Answer 1

$\theta=atan\frac{SO}{SP}=17.84°,$

$r=OP=\sqrt{OS^2+SP^2}=91.3947~m,$

$\dot{r}=v_r=v cos \theta=37.1247~\frac ms,$

$v_{\theta}=-v sin \theta=-11.9480~\frac ms,$

$v_{\theta}=r\dot{\theta}\implies \dot{\theta}=\frac{v_{\theta}}{r}=-0.1307~\frac ms,$

$a_r=acos \theta=9.6143~\frac{m}{s^2},$

$a_{\theta}=-asin \theta=-3.0942~\frac{m}{s^2},$

$a_r=\ddot{r}-r\dot{\theta}^2,\implies \ddot{r}=a_r+r\dot{\theta}^2=11.176~\frac{m}{s^2},$

$a_{\theta}=r\ddot{\theta}+2\dot{r}\dot{\theta}\implies \ddot{\theta}=\frac{a_{\theta}-2\dot{r}\dot{\theta}}{r}=-0.0723~\frac{rad}{s^2}.$