Suppose the position of a particle (in metres) on a number...

Suppose the position of a particle (in metres) on a number line is given by $f(t) = x^3$ for $t \in [0, 5]$ measured in seconds. The distance travelled by the particle over this time period is The velocity of the particle at $t = 3$ is The acceleration of the particle at $t = 1$ is

Transcript

00:01 In this question, we are given the function acceleration of a particle and we know that the velocity of the particle at t equals 2 equals to 3 and we are asked to find the difference between the position functions at t equals 5 and t equals 3.

00:16 So first of all, recall that the velocity is equal to the, sorry, acceleration equals to the derivative of velocity.

00:32 Therefore, velocity function equals to the integral of the acceleration function, equals to the integral of 3 minus 4 t d t, and this equals to 3t minus 4 t squared over 2.

00:53 So 4 and 2 cancel, therefore we are going to get minus 2 t squared plus the constant of integration c.

00:59 To find the constant of integration, we are going to use the initial condition v of 2 equals 3.

01:06 Now plug in t equals 2 in the expression that we found using the integral.

01:13 So we're going to get 3 times 2 minus 2 times 2 squared plus c.

01:20 This gives us 3 equals 6 minus 4 times 2 is 8, minus 8 plus c.

01:28 And this equals to negative 2.

01:31 Moving it to the left hand side gives us 5 equals c.

01:34 C equals 5 therefore v of t equals to 3t minus 2 t squared plus 5 finally since s of t v of t is the integral of s of t is the derivative of t this implies that s of t equals to the integral of v of t which equals to the integral of t which equals to the integral of 3t minus 2 t squared plus 5.

02:28 This in turn equals to 3 halves t squared minus 2 thirds t cube plus 5t plus some new constant of integration, let's say c1...

Question

Suppose the position of a particle (in metres) on a number line is given by $f(t) = x^3$ for $t \in [0, 5]$ measured in seconds. The distance travelled by the particle over this time period is The velocity of the particle at $t = 3$ is The acceleration of the particle at $t = 1$ is

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