The velocity-time graph of a particle in one-dimensional motion is shown in Which of the following formulae are correct for describing the motion of the particle over the time-interval t (a) x(t₂ ) = x(t₁) + v (t1) (t₂ – t₁) +(½) a (t₂ – t₁)₂ (b) v(t₂ ) = v(t₁) + a (t₂ – t₁) (c) vaverage = (x(t₂) – x(t1))/(t₂ – t₁) (d) aaverage = (v(t₂) – v(t1))/(t₂ – t₁) (e) x(t₂ ) = x(t₁) + vaverage (t₂ – t₁) + (½) aaverage (t₂ – t₁)₂ (f) x(t₂) – x(t₁) = area under the v-t curve bounded by the t-axis and the dotted line shown.
The given graph is shown below:
The equations are given as,
(a) x ( t 2 ) = x ( t 1 ) + v ( t 1 ) ( t 1 − t 2 ) + 1 2 a ( t 2 − t 1 ) 2
(b) v ( t 2 ) = v ( t 1 ) + a ( t 2 − t 1 )
(c) v average = x ( t 2 ) − x ( t 1 ) ( t 2 − t 1 )
(d) a average = v ( t 2 ) − v ( t 1 ) ( t 2 − t 1 )
(e) x ( t 2 ) = x ( t 1 ) + v average ( t 1 − t 2 ) + 1 2 a average ( t 2 − t 1 ) 2
(f) x ( t 2 ) − x ( t 1 ) = area under the v − t curve bounded by the t − axis and the dotted line shown
The equations given in the options (a), (b) and (e), are of straight lines.
From the given graph, it is observed that the graph between t 1 and t 2 has a non-uniform slope. The motion in this interval cannot be a straight line. Therefore, the equations given in options (a), (b) and (e) do not define the motion of the particle over the time interval t 1 and t 2 .
The average velocity is the ratio of the difference in the displacement at two positions to the difference in the time interval.
The average velocity between time t 2 and t 1 is given as,
v average = x ( t 2 ) − x ( t 1 ) ( t 2 − t 1 )
The average acceleration is the ratio of the difference in the velocity at two positions to the difference in the time interval.
The average acceleration between time t 2 and t 1 is given as,
a average = v ( t 2 ) − v ( t 1 ) ( t 2 − t 1 )
The area under the velocity-time graph represents the displacement of the body between two time intervals. This is because displacement is the product of the velocity and the respective time.
The displacement of the body between time t 2 and t 1 is given as,
x ( t 2 ) − x ( t 1 ) = area under the v − t curve bounded by the t − axis and the dotted line
Thus, the correct equations are (c), (d) and (f).
The given graph is shown below:
The equations are given as,
(a)
(b)
(c)
(d)
(e)
(f)
The equations given in the options (a), (b) and (e), are of straight lines.
From the given graph, it is observed that the graph between
The average velocity is the ratio of the difference in the displacement at two positions to the difference in the time interval.
The average velocity between time
The average acceleration is the ratio of the difference in the velocity at two positions to the difference in the time interval.
The average acceleration between time
The area under the velocity-time graph represents the displacement of the body between two time intervals. This is because displacement is the product of the velocity and the respective time.
The displacement of the body between time
Thus, the correct equations are (c), (d) and (f).
)a(t2–t1)2