Question

Problem 1.42 Recall from , that Robbie's position function is R ( t ) = t 3 + 2. What is Robbie's average velocity from time t = 3 till time t = 4 ? What is Robbie's average velocity from time t = 3 till time t = 3.5 ? What is Robbie's average velocity from time t = 3 till time t = 3.1 ? What is Robbie's instantaneous velocity at time t=3?

Answer 1

3 in each case.

Step-by-step explanation:

We've got the position function
`R(t)`. We can see that it's the equation of a line. That is, if we were to graph the position R against the time t, we would get a line.

On the other hand, the velocity is the rate of change of the position related to the time. Since the position is a lineal function, we can calculate the speed taking
`t_0` as the starting moment and
`t_1` as the end moment measure the distance travelled. The distance travelled from instant 0 to instant 1 would the be
`R(t_1)-R(t_0)` and the time spend in that time would be
`t_1-t_0`. So the velocity v is:

`v=(R(t_1)-R(t_0))/(t_1-t_0)\\v=(t_1*3+2-(t_0*3+2))/(t_1-t_0)\\v=(3t_1+2-3t_0-2)/(t_1-t_0)\\v=(3(t_1-t_0))/(t_1-t_0)\\v=3`

So we get that the velocity is the same for whichever two points we use to calculate it. Hence, it's constant and it's value is 3, so the average is also the same no matter the time we use.

The units are not shown here because the question doesn't say which are used, but in the metric system it's meters/seconds.

Answer 2

a)
`v_(ave)=37(m)/(s)`

b)
`v_(ave)= 31.75(m)/(s)`

c)
`v_(ave)= 27.91(m)/(s)`

d)
`v(3)=27(m)/(s)`

Step-by-step explanation:

Robbie's position function is
`R(t)=t^3+2`.

a) Average velocity from time t = 3 till time t = 4 will be:

`v_(ave)=(x_f-x_i)/(t_f-t_i) =(R(4)-R(3))/(4s-3s)=(66m-29m)/(1s)= 37(m)/(s)`

b) Average velocity from time t = 3 till time t = 3.5 will be:

`v_(ave)=(x_f-x_i)/(t_f-t_i) =(R(3.5)-R(3))/(3.5s-3s)=(44.875m-29m)/(0.5s)= 31.75(m)/(s)`

c) Average velocity from time t = 3 till time t = 3.1 will be:

`v_(ave)=(x_f-x_i)/(t_f-t_i) =(R(3.1)-R(3))/(3.1s-3s)=(31.79m-29m)/(0.1s)= 27.91(m)/(s)`

d) Instantaneous velocity at time t=3 is:

`v(3)= \lim_(\Delta t \to 0) (R(3+\Delta t )-R(3))/(\Delta t) = \lim_(\Delta t \to 0) ((3+\Delta t )^3+2-3^3-2)/(\Delta t)`

⇒
`\lim_(\Delta t \to 0) (3^3+3^3\Delta t+3^2\Delta t^2+\Delta t^3-3^3)/(\Delta t) = \lim_(\Delta t \to 0) ((27+9\Delta t+\Delta t^2)\Delta t)/(\Delta t)`

⇒
`\lim_(\Delta t \to 0) 27+9\Delta t+\Delta t^2=27(m)/(s)`

∴
`v(3)=27(m)/(s)`