Question

A particle is projected with velocity of 10 m/s in a resistive medium which offers retardation given by a = 5v. If velocity (in m/s) of particle as function of time t is given as v= 10e-kt then find k.

help​

Answer 1

Answer:

Given : v

o

=10 m/s θ

o

=60

o

Angle of inclination α=30

o

Time of flight T= g cosα

2v

o

sinθ

o

⟹ T= 10×cos30

o

2×10×sin60

o

=2 s

Answer 2

The value of 'k' is equal to -5.

Given:

Initial velocity of the particle (u) = 10 m/s

Retardation: a = 5v

$Velocity \:of \:the\: body\:(v)=10e^{-kt}$

To Find:

The value of 'k'.

Solution:

→ The acceleration of a particle is defined as the rate of change of velocity.

The acceleration can be calculated by dividing the total change in velocity (Δv) by the total time taken (Δt).

                                           $\therefore a=\frac{\triangle v}{\triangle t}$

→ In the differential form: The acceleration (a) of body is equal to the differentiation of the velocity (v) of the body with respect to time (t).

                                           $\therefore a=\frac{dv}{dt}$

→ Hence calculating the value of acceleration (a) using the differential form:

   $\therefore a=\frac{dv}{dt} \\\\\therefore a=\frac{d}{dt}(10e^{-kt} )\\\\\therefore a=10(-k)e^{-kt} \\\\\therefore a =-10k(e^{-kt})$

→ As we know a = 5v, therefore putting the values of 'a' and 'v' in the equation:

    $\because a=5v$

    $\therefore -10k(e^{-kt}) =5(10e^{-kt})\\\\\therefore -10k=50$

        $\therefore k=-5$

Therefore the value of 'k' comes out to be equal to -5.

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