Home
Class 12
PHYSICS
The acceleration of a particle which sta...

The acceleration of a particle which starts from rest varies with time according to relation a=2t+3. The velocity of the particle at time t would be

A

`2t^3+3`

B

`t^2+3t`

C

`2t^2+3t`

D

zero

Text Solution

AI Generated Solution

The correct Answer is:
To find the velocity of a particle whose acceleration varies with time according to the relation \( a = 2t + 3 \), we can follow these steps: ### Step 1: Understand the relationship between acceleration and velocity Acceleration \( a \) is defined as the rate of change of velocity with respect to time. Mathematically, this is expressed as: \[ a = \frac{dv}{dt} \] ### Step 2: Substitute the given acceleration into the equation From the problem, we have: \[ a = 2t + 3 \] So we can write: \[ \frac{dv}{dt} = 2t + 3 \] ### Step 3: Rearrange the equation for integration We can rearrange this equation to isolate \( dv \): \[ dv = (2t + 3) dt \] ### Step 4: Integrate both sides Now we integrate both sides. The left side integrates with respect to \( v \) and the right side with respect to \( t \): \[ \int dv = \int (2t + 3) dt \] The left side gives: \[ v \] The right side can be integrated as follows: \[ \int (2t + 3) dt = \int 2t \, dt + \int 3 \, dt = t^2 + 3t + C \] where \( C \) is the constant of integration. ### Step 5: Determine the constant of integration Since the particle starts from rest, we know that at \( t = 0 \), the velocity \( v = 0 \): \[ v(0) = 0 = 0^2 + 3(0) + C \implies C = 0 \] ### Step 6: Write the final expression for velocity Substituting \( C \) back into the equation gives us: \[ v = t^2 + 3t \] ### Conclusion Thus, the velocity of the particle at time \( t \) is: \[ v(t) = t^2 + 3t \] ---
Doubtnut Promotions Banner Mobile Dark
|

Topper's Solved these Questions

  • MECHANICAL PROPERTIES OF SOLIDS

    AAKASH INSTITUTE ENGLISH|Exercise Assignment(Section-D)|14 Videos
  • MOCK TEST 10

    AAKASH INSTITUTE ENGLISH|Exercise Elxample|30 Videos

Similar Questions

Explore conceptually related problems

The acceleration of a'particle starting from rest, varies with time according to the relation a=kt+c . The velocity of the particle after time t will be :

The acceleration a of a particle starting from rest varies with time according to relation, a=alphat+beta . Find the velocity of the particle at time instant t. Strategy : a=(dv)/(dt)

A particle starts from rest and moves with an acceleration of a={2+|t-2|}m//s^(2) The velocity of the particle at t=4 sec is

A particle starts from rest and moves with an acceleration of a={2+|t-2|}m//s^(2) The velocity of the particle at t=4 sec is

Starting from rest, the acceleration of a particle is a=2(t-1). The velocity of the particle at t=5s is :-

The acceleration of a particle varies with time t seconds according to the relation a=6t+6ms^-2 . Find velocity and position as funcitons of time. It is given that the particle starts from origin at t=0 with velocity 2ms^-1 .

A particle starts from rest at t=0 and moves in a straight line with an acceleration as shown below. The velocity of the particle at t=3 s is

Tangental acceleration of a particle moving in a circle of radius 1 m varies with time t as shown in figure (initial velocity of the particle is zero). Time after which total acceleration of particle makes an angle of 30^(@) with radial acceleration is

The acceleratin of a particle increases linearly with time t as 6t. If the initial velocity of the particles is zero and the particle starts from the origin, then the distance traveled by the particle in time t will be

The position x of a particle varies with time t according to the relation x=t^3+3t^2+2t . Find the velocity and acceleration as functions of time.