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A point traversed half a circle of radius R=160cm during time interval tau=10.0s. Calculate the following quantities averaged over that time: (a) the mean velocity ltlt v gtgt, (b) the modulus of the mean velocity vector |ltlt v gtgt|, (c) the modulus of the mean vector of the total acceleration |ltlt w gtgt| if the point moved with constant tangent acceleration. |
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Answer» `LT v gt = ("Total distance COVERED")/("Time elapsed")` `=s/t=(piR)/(tau)=50cm//s` (1) (b) Modulus of mean velocity VECTOR `|lt vecv gt|=(|Deltavecr|)/(Deltat)=(2R)/(tau)=32cm//s` (2) (c) Let the point moves from i to f along the half circle (figure) and `v_0` and `v` be the spe at the points respectively. We have `(dv)/(dt)=w_t` or, `v=v_0+w_t t` (as `w_t` is constant, according to the problem) So, `lt v gt =(underset0oversettint(v_0+w_t t)dt)/(underset0oversettintdt)=(v_0+(v_0+w_t t))/(2)=(v_0+v)/(2)` (3) So, from (1) and (3) `(v_0+v)/(2)=(piR)/(tau)` (4) Now the modulus of the mean vector of total acceleration `|lt vecw gt|=(|Deltavecv|)/(Deltat)=(|vecv-vecv_0|)/(tau)=(v_0+v)/(tau)` (see Figure) (5) Using (4) and (5), we get: `|lt vecw gt|=(2piR)/(tau^2)` 
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