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Motion in a Straight Line

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Question : 23 of 27
Marks: +1, -0
A three-wheeler starts from rest, accelerates uniformly with 1 m s−2^{-2}on a straight road for 10 s, and then moves with uniform velocity. Plot the distance covered by the vehicle during the nth second (n = 1, 2, 3..) versus n. What do you expect this plot to be during accelerated motion : a straight line or a parabola?
Solution:  
Initial velocity, u=0, a=1 m s−2, t=10 su = 0,\ a = 1\ \mathrm{m\,s}^{-2},\ t = 10\ \mathrm{s} , If SnS_n be the distance covered by the three-wheeler in nth second. Then
Sn=u+a2(2n−1),S_{n}=u+\frac{a}{2}(2n-1),
Sn=a2(2n−1)=12(2n−1)S_{n}=\frac{a}{2}(2n-1)=\frac{1}{2}(2n-1)
For n=1,n=1,
S1=12(2×1−1)=0.5 mS_{1}=\frac{1}{2}(2\times 1-1)=0.5\ \mathrm{m}
n=2,n=2,
S2=12(2×2−1)=1.5 mS_{2}=\frac{1}{2}(2\times 2-1)=1.5\ \mathrm{m}
n=3,n=3,
S3=12(2×3−1)=2.5 mS_{3}=\frac{1}{2}(2\times 3-1)=2.5\ \mathrm{m} and so on
The following is the table for the distance SnthS_n^{\text{th}} travelled in nthn_{\text{th}} second.
  n(s)n(s)  1  2  3  4  5  6  7  8  9  10
  Sn(m)S_n(m)  0.5  1.5  2.5  3.5  4.5  5.5   6.5  7.5  8.5  9.5
Also Dn=D_n= distance covered by the particle in 10 sec.
Dn=un+12an2D_{n}=u n+\frac{1}{2}a n^{2}
=0×10+12×1×102=0\times 10+\frac{1}{2}\times 1\times 10^{2}
(n=10)(n=10)
=50 m.= 50\ \mathrm{m}.
As the equation describing the relation between DnD_n and n is of seconddegree, so the graph of the vehicle from start to 10th^{\text{th}}second is expectedto be a parabola. Velocity of the vehicle at the end of 10th^{\text{th}} second isv=0+1×10=10 m s−1v = 0 + 1 \times 10 = 10\ \mathrm{m\,s}^{-1} and it will move with this velocity after 10 seconds.Thus the graph will be a straight line inclined to time axis for uniformlyaccelerated motion. The plot of the distance covered by the three wheelerwith time is shown in the graph.
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