Question

A particle is moving on a line, where its position S in meters is a function of time t in seconds given by S = t³ + at² + bt + c where a, b, c are constant. It is known that at t = 1 seconds, the position of the particle is given by S = 7 m. Velocity is 7 m/s and acceleration is 12 m/s². The values of a, b, c are ______.

Options

• –3, 2, 7

• 3, –2, 5

• 3, 2, 1

• –3, 2, –1

Answer 1

A particle is moving on a line, where its position S in meters is a function of time t in seconds given by S = t³ + at² + bt + c where a, b, c are constant. It is known that at t = 1 seconds, the position of the particle is given by S = 7 m. Velocity is 7 m/s and acceleration is 12 m/s². The values of a, b, c are 3, –2, 5.

Explanation:

S = t³ + at² + bt + c

At t = 1 second,

Position = 7 m

Velocity = 7 m/s

Acceleration = 12 m/s²

At t = 1 s

and S = 7 m

1 + a + b + c = 7

⇒ a + b + c = 6  ...(1)

Differentiate equation,

S = t³ + at² + bt + c

w.r.t.t, we get

`("ds")/("dt")` = velocity, 3t² + 2at + b

At t = 1 s,

V = 7 m/s

3 + 2a + b = 7

⇒ 2a + b = 4  ...(2)

Differentiating equation,

`("ds")/("dt")` = 3t² + 2at + b

w.r.t.t, we get

`("d"^2"s")/("dt"^2)` = acceleration = 6t + 2a

At t = 1 s,

Acceleration = 12 m/s²

6.(1) + 2a = 12

⇒ 2a = 6

⇒ a = 3

b = 4 – 2a

= 4 – 6 = –2

From equation (1) a + b + c = 6

3 – 2 + c = 6

⇒ c = 5

(a, b, c) = (3, –2, 5)