All Problems

Kinematics

Problem 1.21

At the moment \(t=0\) a particle leaves the origin and moves in the positive direction of the \(x\) axis. Its velocity varies with time as \(v=v_{0}(1-t / \tau),\) where \(v_{0}\) is the initial velocity vector whose modulus equals \(v_{0}=10.0 \mathrm{~cm} / \mathrm{s} ; \tau=5.0 \mathrm{~s}\). Find: (a) the \(x\) coordinate of the particle at the moments of time 6.0 , \(10,\) and \(20 \mathrm{~s}\) (b) the moments of time when the particle is at the distance \(10.0 \mathrm{~cm}\) from the origin; (c) the distance \(s\) covered by the particle during the first 4.0 and \(8.0 \mathrm{~s} ;\) draw the approximate plot \(s(t)\)

Reveal Answer
 (a) x=v0t(1−t/2τ),x=0.24,0 and −4.0 m (b) 1.1,9 and 11 s24 and 34 cm respectively.  (c) s={(1−t/2τ)v0t for t⩽τ[1+(1−t/τ)2]v0t/2 for t⩾τ\text { (a) } x=v_{0} t(1-t / 2 \tau), x=0.24,0 \text { and }-4.0 \mathrm{~m} \text { (b) } 1.1,9 \text { and } 11 \mathrm{~s} 24 \text { and } 34 \mathrm{~cm} \text { respectively. } \text { (c) } s=\left\{\begin{array}{l} (1-t / 2 \tau) v_{0} t \text { for } t \leqslant \tau \\ {\left[1+(1-t / \tau)^{2}\right] v_{0} t / 2 \quad \text { for } \quad t \geqslant \tau} \end{array}\right.
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