A particle of mass 2 kg is moving along the x-axis. There is no force ...
Given:
- Mass of the particle (m) = 2 kg
- Force acting on the particle (F(t)) = 6t^2 N/s^2
- Initial velocity of the particle at t = 0 (v0) = 1000 m/s
- Time interval between t = 0 sec and t = 5 sec when force is acting on the particle
To find:
Velocity of the particle at t = 10 sec (v)
Approach:
To find the velocity of the particle at t = 10 sec, we need to calculate the total change in momentum of the particle over the time interval between t = 0 sec and t = 10 sec. Since there is no force acting on the particle after t = 5 sec, the momentum of the particle remains constant from t = 5 sec to t = 10 sec.
Step 1: Calculate the change in momentum from t = 0 sec to t = 5 sec
- The force acting on the particle is given by F(t) = 6t^2 N/s^2.
- The acceleration of the particle (a) can be calculated using Newton's second law, F = ma.
- Integrating the acceleration with respect to time will give the change in velocity (Δv).
- Integrating the change in velocity with respect to time will give the change in momentum (Δp).
Step 2: Calculate the change in momentum from t = 5 sec to t = 10 sec
- Since there is no force acting on the particle after t = 5 sec, the momentum remains constant. Therefore, the change in momentum is zero.
Step 3: Calculate the final velocity at t = 10 sec
- The final momentum at t = 10 sec is the sum of the initial momentum and the change in momentum from t = 0 sec to t = 5 sec.
- The final velocity can be calculated by dividing the final momentum by the mass of the particle.
Calculation:
Step 1:
- The acceleration of the particle is given by a = F(t)/m = (6t^2)/2 = 3t^2 m/s^2.
- Integrating the acceleration, we get the change in velocity as Δv = ∫(3t^2)dt = t^3 m/s.
- Integrating the change in velocity, we get the change in momentum as Δp = ∫(t^3)dt = (1/4)t^4 kg.m/s.
Step 2:
- There is no force acting on the particle after t = 5 sec, so the change in momentum is zero, i.e., Δp = 0 kg.m/s.
Step 3:
- The final momentum at t = 10 sec is the sum of the initial momentum and the change in momentum from t = 0 sec to t = 5 sec, i.e., p = m*v0 + Δp.
- The final velocity at t = 10 sec is given by v = p/m.
Substituting the values:
- Δp = (1/4)*(5^4) = 625/4 kg.m/s
- p