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Average Velocity in 2D Motion

For any arbit...

Question

For any arbitrary motion in space, which of the following relations are true : (a) v _average = (1/2) (v (t₁) + v (t₂)) (b) v _average = [r(t₂) - r(t₁) ] /(t₂ – t₁) (c) v (t) = v (0) + a t (d) r (t) = r (0) + v (0) t + (1/2) a t₂ (e) a _average =[ v (t2) - v (t₁ )] /( t₂ – t₁) (The ‘average’ stands for average of the quantity over the time interval t₁to t2)

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Solution

a)

The given relation is,

v average =( 1 2 )( v( t 1 )+v( t 2 ) )

Since, the motion is arbitrary in space therefore, the average velocity cannot be given by this relation. This relation holds good for uniformly accelerated motion.

Thus, for arbitrary motion in space, the given relation is false.

b)

The given relation is,

v average = ( r( t 2 )−r( t 1 ) ) ( t 2 − t 1 )

Any arbitrary motion in space can be represented by the given relation.

Thus, the given relation is true for arbitrary motion.

c)

The given relation is,

v( t )=v( 0 )+at

The given relation is valid only when a particle accelerating uniformly.

Thus, for arbitrary motion in space, the given relation is false.

d)

The given relation is,

r( t )=r( 0 )+v( 0 )t+( 1 2 )a t 2

For an arbitrary motion, the acceleration of the particle may not be uniform.

Thus, the given relation is false for arbitrary motion.

e)

The given relation is,

a average = ( v( t 2 )−v( t 1 ) ) ( t 2 − t 1 )

Any arbitrary motion in space can be represented by the given relation.

Thus, the given relation is true for arbitrary motion.

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v_{a v e r a g e} = (1 / 2) (v (t_{1}) + v (t_{2}))

v_{a v e r a g e} = [r (t_{2}) - r (t_{1})] / (t_{2} - t_{1})

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r (t) = r (0) + v (0) t + (1 / 2) a

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a_{a v e r a g e} = [v (t_{2}) - v (t_{1})] / (t_{2} - t_{1})

t_{1}

t_{2}

t_{1}

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For any arbitrary motion in space, which of the following relations are true:

(a)

(b)

(c)

(d)

(e)

(The ‘average’ stands for average of the quantity over the time interval t₁ to t₂)

The velocity-time graph of a particle in one-dimensional motion is shown in Fig. 3.29:

Which of the following formulae are correct for describing the motion of the particle over the time-interval t₂ to t₁?

(a) x(t₂) = x(t₁) + v(t₁)(t₂–t₁) + ()a(t₂–t₁)²

(b) v(t₂)= v(t₁) + a(t₂–t₁)

(d) a_Average = (v(t₂) – v(t₁)) / (t₂ – t₁)

(e) x(t₂) = x(t₁) + v_Average(t₂ – t₁) + ()a_Average(t₂ – t₁)²

(f) x(t₂) – x(t₁) = area under the v–t curve bounded by the t-axis and the dotted line shown.

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