Question

For any arbitrary motion in space, which of the following relations are true: a. $V_{average}= (\frac{1}{2})[v(t_1)+v(t_2)]$ b. $V_{average}=\frac {[r(t_2)-r(t_1)]} {(t_2-t_1)}$ c. v (t) = v (0) + at d. r (t) = r (0) + v (0) t + (1/2) at² e. $a_{average}=\frac {[v(t_2)-v(t_1)]} {(t_2-t_1)}$ (The 'average' stands for average of the quantity over the time interval $t_1 to t_2$)

Answer 1

(a) according to question, motion of particle in space is arbitrary, but this one equation shows a special case of motion. Therefore, arbitrarily the average velocity of the particle cannot be given by this equation.

(b) this equation is taken from definition of average velocity and it is applicable in everywhere to find average acceleration , therefore arbitrary motion of the particle can be represented by this equation.

(c) The motion of the particle is arbitrary. The acceleration of the particle may also be non-uniform. Hence, this equation cannot represent the motion of the particle in space.

(d) The motion of the particle is arbitrary; acceleration of the particle may also be non-uniform. Hence, this equation cannot represent the motion of particle in space.

(e) this equation is taken from definition of average acceleration and it is applicable in everywhere to find average acceleration . Therefore, arbitrary motion of the particle can be represented by this equation.

Answer 2

For any arbitrary motion in space, which of the following relations are true: a. $V_{average}= (\frac{1}{2})[v(t_1)+v(t_2)]$ b. $V_{average}=\frac {[r(t_2)-r(t_1)]} {(t_2-t_1)}$ c. v (t) = v (0) + at d. r (t) = r (0) + v (0) t + (1/2) at² e. $a_{average}=\frac {[v(t_2)-v(t_1)]} {(t_2-t_1)}$ (The 'average' stands for average of the quantity over the time interval $t_1 to t_2$)Which of the following functions of time represent (a) simple harmonic, (b) periodic but not simple harmonic and (c) non periodic motion? Give period for each case of periodic motion (ω is any positive constant) a. sign ωt - cos ωt b. sin² ωt c. 3 Cos (π/4 - ωt) d. cos ωt + cos 3ωt + cos 5ωt e. exp(-ω²t²) f. 1 + ωt + ω²t²