A particle moving along the x-axis has its velocity described by the function v =2t squared m/s, where t is in s. Its initial position is x = 2.8m at t = 0 s. At 2.7s , what is the particle's position?

Question

A particle moving along the x-axis has its velocity described by the function v =2t squared m/s, where t is in s. Its initial position is x = 2.8m at t = 0 s. At 2.7s , what is the particle's position?

Answer 1

v=dx/dt=2t^2 dx=2t^2dt integrating both side gives: x=2/3t^3+c substituting t=0 and x=2.8 gives c=2.8 therefore x=2/3t^3+2.8 so when t=2.7 x=2/3(2.7)^3+2.8=15.9m

Answer 2

To find the particle's position at a specific time, we need to integrate the velocity function with respect to time.

Given that the velocity function is v = 2t^2 m/s, we can integrate it to find the position function.

∫(v) dt = ∫(2t^2) dt = (2/3)t^3 + C

Since the initial position is given as x = 2.8 m at t = 0 s, we can substitute these values into the position function to find the constant of integration, C.

x = (2/3)t^3 + C
2.8 = (2/3)(0)^3 + C
2.8 = C

Now we can substitute the value of C back into the position function:

x = (2/3)t^3 + 2.8

To find the particle's position at t = 2.7 s, we can substitute t = 2.7 into the position function:

x = (2/3)(2.7)^3 + 2.8

Calculating this expression, we find:

x ≈ 22.986 m

Therefore, at 2.7 s, the particle's position is approximately 22.986 m.

Answer 3

To determine the particle's position at a given time, we need to integrate its velocity function. The velocity function v(t) = 2t^2 m/s represents the rate of change of the particle's position with respect to time.

To integrate v(t) with respect to t, we will apply the indefinite integral.

∫v(t) dt = ∫2t^2 dt

Now, let's integrate the function.

∫2t^2 dt = 2 * ∫t^2 dt

Using the power rule of integration, we increase the power of t and divide by the new power:

= 2 * [(t^2+1)/3] + C

Note: C represents the constant of integration and accounts for any initial conditions.

Now, to find the particle's position at 2.7 seconds, we substitute t = 2.7 into the integrated expression:

= 2 * [(2.7^2+1)/3] + C

Evaluating the expression:
= 2 * [(7.29 + 1) / 3] + C
= 2 * (8.29/3) + C
= 5.53 + C

Given that the initial position at t = 0s is x = 2.8m, we can use this information to find the value of C:

2.8 = 5.53 + C

Rearranging the equation:
C = 2.8 - 5.53
C = -2.73 m

Now, substituting the value of C back into our integrated expression:

Position at t = 2.7s = 5.53 - 2.73
Position at t = 2.7s = 2.8 m

Therefore, at 2.7 seconds, the particle's position is 2.8 meters.