Question #177176

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





1
Expert's answer
2021-04-01T18:34:39-0400

θ=atanSOSP=17.84°,\theta=atan\frac{SO}{SP}=17.84°,

r=OP=OS2+SP2=91.3947 m,r=OP=\sqrt{OS^2+SP^2}=91.3947~m,

r˙=vr=vcosθ=37.1247 ms,\dot{r}=v_r=v cos \theta=37.1247~\frac ms,

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

vθ=rθ˙    θ˙=vθr=0.1307 ms,v_{\theta}=r\dot{\theta}\implies \dot{\theta}=\frac{v_{\theta}}{r}=-0.1307~\frac ms,

ar=acosθ=9.6143 ms2,a_r=acos \theta=9.6143~\frac{m}{s^2},

aθ=asinθ=3.0942 ms2,a_{\theta}=-asin \theta=-3.0942~\frac{m}{s^2},


ar=r¨rθ˙2,    r¨=ar+rθ˙2=11.176 ms2,a_r=\ddot{r}-r\dot{\theta}^2,\implies \ddot{r}=a_r+r\dot{\theta}^2=11.176~\frac{m}{s^2},

aθ=rθ¨+2r˙θ˙    θ¨=aθ2r˙θ˙r=0.0723 rads2.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}.


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