A force F = 3t^2 N acts on a particle for 3 s after which the particle moves freely. The particle is initially at the origin with zero velocity. Determine its position at time t=5 s. Assume the mass of the particle is 0.1 kg.

To determine the position of the particle at time t = 5 seconds, we need to analyze the motion of the particle under the influence of the force acting on it for the first 3 seconds, and then its subsequent free motion. Let's break this down step by step.

Step 1: Analyze the Force

The force acting on the particle is given by:

F(t) = 3t^2 N

This force is not constant; it varies with time. We can calculate the acceleration of the particle by using Newton's second law, which states:

F = ma

Where:

• m is the mass of the particle (0.1 kg)
• a is the acceleration

Rearranging this gives us:

a(t) = F(t) / m

Substituting in the values:

a(t) = (3t^2) / 0.1 = 30t^2 m/s²

Step 2: Determine the Velocity

To find the velocity of the particle, we need to integrate the acceleration over time. The velocity v(t) can be expressed as:

v(t) = ∫a(t) dt

Substituting in our expression for acceleration:

v(t) = ∫30t^2 dt = 10t^3 + C

Since the particle starts from rest (initial velocity is 0 at t = 0), we can determine the constant C:

v(0) = 10(0)^3 + C = 0 ⟹ C = 0

This means:

v(t) = 10t^3

Step 3: Find the Position

Next, we need to find the position of the particle. The position x(t) can be calculated by integrating the velocity function:

x(t) = ∫v(t) dt

Substituting our expression for velocity:

x(t) = ∫10t^3 dt = 2.5t^4 + D

Again, since the particle starts at the origin (x = 0 when t = 0), we can find D:

x(0) = 2.5(0)^4 + D = 0 ⟹ D = 0

This gives us:

x(t) = 2.5t^4

Step 4: Calculate Position at t = 3 s

Now, let's find the position at t = 3 seconds, the end of the force application:

x(3) = 2.5(3)^4 = 2.5 × 81 = 202.5 m

At this point, we also need to determine the velocity at t = 3 seconds:

v(3) = 10(3)^3 = 10 × 27 = 270 m/s

Step 5: Free Motion from t = 3 s to t = 5 s

After t = 3 seconds, the particle will move with a constant velocity of 270 m/s since no forces are acting on it. To find the position at t = 5 seconds, we need to calculate the additional distance traveled during this time:

Time interval during free motion:

Δt = 5 s - 3 s = 2 s

Distance traveled during free motion:

d = v × Δt = 270 m/s × 2 s = 540 m

Final Position Calculation

The total position at t = 5 seconds is the initial position at t = 3 seconds plus the distance traveled during the free motion:

x(5) = x(3) + d = 202.5 m + 540 m = 742.5 m

Thus, the position of the particle at time t = 5 seconds is 742.5 meters.