Question

The displacement (in meter) of a particle moving along x-axis is given by x= 18+t^2. Calculate instantaneous velocity at t=2 second , average velocity at t=3 second and instantaneous acceleration at t=2 second, t=3 second.

Answer 1

Solution :

Displacement of the particle w.r.t time is given as : 18 + t².

Differentiating x w.r.t time, we get velocity :

$\longrightarrow$ v = dx/dt

$\longrightarrow$ v = d(18 + t²)/dt

$\longrightarrow$ v = 2t m/s

Instantaneous Velocity at t = 2s, v = 2(2) = 4 m/s

Average Velocity :

$\sf \overline{v} = \dfrac{ \Delta{x}}{\Delta{t}}$

Now,

$\longrightarrow$ v = [(18 + 3²) - (18 + 0²)]/3 - 0

$\longrightarrow$ v = 9/3

$\longrightarrow$ v = 3 m/s

Differentiating v w.r.t t, we get acceleration :

$\longrightarrow$ a = dv/dt

$\longrightarrow$ a = d(2t)/dt

$\longrightarrow$ a = 2m/s²

At every instant of time, instantaneous acceleration is 2m/s².

Answer 2

Given:-

• Displacement of a particle moving along x-axis is given by, x = 18 + t²

To find:-

• Instantaneous velocity at t = 2 sec
• Average velocity at t = 3 sec
• Instantaneous acceleration at t=2 sec, t = 3 sec

Knowledge required:-

• Instantaneous velocity is given by the differential coefficient position of the body with respect to time.

         v(t) = d(x)/d(t)

• Instantaneous acceleration is given by the coefficient of the velocity of the body with respect to time.  

         a(t) = d(v)/d(t)

• Average velocity is defined as the change in position divided by the time interval for displacement.

         v' = Δx / Δt  

Solution:-

Let us find the velocity of the particle at any instant 't'

→ v(t) = d(x)/d(t)

→ v(t) = d(18 + t²)/d(t)

→ v(t) = 2t

So,

The Instantaneous velocity of the particle at t=2 sec

→ v(2) = 2(2) = 4 m/s

Now, Calculating the average velocity of a particle at t=3 sec

→ v' = Δx/Δt

→ v' = [( 18 + (3)²) - (18+(0)²)]/(3 - 0)

→ v' = (27-18)/(3)

→ v' = 3 m/s

Further,

Let us find the acceleration of the particle at any instant 't'

→ a(t) = d(v)/d(t)

→ a(t) = d(2t)/d(t)

→ a(t) = 2 m/s²

So, acceleration of the particle at any instant of time will be 2 m/s².

Therefore,

• Instaneous velocity of particle at t=2sec will be 4 m/s
• Average velocity of particle at t=3sec will be 3 m/s
• Instantaneous acceleration of particle at t=2sec and t=3sec will be 2 m/s².