A particle moves on the x-axis with an acceleration, a=(6t-4)m/s2. Find the position and

velocity of the particle at t=3 , if the particle is at origin and has a velocity of
10 m/s
ms when
t=0.

hahaha.

Soalan assignment~!!!

To find the position and velocity of the particle at t=3, we'll need to integrate the equation for acceleration with respect to time.

Given: acceleration, a = (6t - 4) m/s^2
The velocity can be found by integrating the acceleration with respect to time:
v = ∫ a dt = ∫ (6t - 4) dt

To find the position, integrate the velocity with respect to time:
s = ∫ v dt

First, let's find the velocity at t=3:
To integrate 6t - 4, we can split it into two separate integrals:
v = ∫ (6t - 4) dt = ∫ 6t dt - ∫ 4 dt

Now, let's solve each integral separately:
∫ 6t dt = 3t^2 + C1, where C1 is the constant of integration
∫ 4 dt = 4t + C2, where C2 is the constant of integration

Combining the two integrals, we get:
v = 3t^2 + C1 - 4t + C2

We know that when t=0, v=10 m/s. Substituting these values into the equation, we can solve for the constants C1 and C2:
10 = 3(0)^2 + C1 - 4(0) + C2
10 = C1 + C2
C1 + C2 = 10 (Equation 1)

Now, let's calculate the position at t=3:
∫ v dt = ∫ (3t^2 + C1 - 4t + C2) dt
s = t^3 + C1t - 2t^2 + C2t + C3, where C3 is the constant of integration

We don't have a specific value for the initial position, but we know that the particle is at the origin when t=0. Therefore, s=0 when t=0. Substituting these values into the equation, we get:
0 = (0)^3 + C1(0) - 2(0)^2 + C2(0) + C3
0 = C3
C3 = 0

Now we have the complete equation for position:
s = t^3 + C1t - 2t^2 + C2t

To find the position at t=3, we substitute t=3 into the equation and solve for C1 and C2:
s = (3)^3 + C1(3) - 2(3)^2 + C2(3)

We also know that the position at t=3 is unknown. Let's denote it as s3. Substituting this into the equation, we get:
s3 = 27 + 3C1 - 18 + 3C2

Now, let's solve Equation 1 for C1 in terms of C2:
C1 = 10 - C2 (Equation 2)

Substituting Equation 2 into the equation for s3, we get:
s3 = 27 + 3(10 - C2) - 18 + 3C2
s3 = 57 + 3C2 - 18 + 3C2
s3 = 3C2 + 39

Therefore, the position at t=3 is given by s3 = 3C2 + 39.

To find the velocity and position at t=3, we need to know the specific value of either C1 or C2.