Question

The equation gives the position s = f(t) of a body moving

on a coordinate line (s in meters, t in seconds).

1) s = 1 + 9(cos t)
Find the body's speed at time t=pi/3 sec.

2) s = 12(sin t) - (cos t)
Find the body's velocity at time t=pi/4 sec.

Answer 1

1) The velocity at this time is of
`-9(√(3))/(2)` meters per second.

2) The velocity at this time is of
`13(√(2))/(2)` meters per second.

Explanation:

This question involves concepts of derivatives.

The velocity is the derivative of the position.

We use these following derivatives:

`(sin(x))^(\prime) = cos(x)`

`(cos(x))^(\prime) = -sin(x)`

1) s = 1 + 9(cos t)

Find the body's speed at time t=pi/3 sec.

We have to find the derivative at
`t = (\pi)/(3)`. So

`v = (1 + 9cos(t))^(\prime) = -9sin(t)`

`v((\pi)/(3)) = -9\sin{(\pi)/(3)}`

`(\pi)/(3)` is a common angle, which has a sine of
`(√(3))/(2)`. So

`-9\sin{(\pi)/(3)} = -9(√(3))/(2)`

The velocity at this time is of
`-9(√(3))/(2)` meters per second.

2) s = 12(sin t) - (cos t)

Find the body's velocity at time t=pi/4 sec.

We have to find the derivative at
`t = (\pi)/(4)`. So

`v = (12sin(t) - cos(t))^(\prime) = 12cos(t) + sin(t)`

`(\pi)/(4)` is a common angle, which has both sine and cosine of
`(√(2))/(2)`. So

`12\cos{(\pi)/(4)} + \sin{(\pi)/(4)} = 12(√(2))/(2) + (√(2))/(2) = 13(√(2))/(2)`

The velocity at this time is of
`13(√(2))/(2)` meters per second.