Question

The position function of a particle is x(t) = 30t −5t 2

.

a. Find the velocity function x˙(t).

b. Find the velocity when t = 2.

c. Find the velocity when t = 4.

d. When is the particle stationary?​

Answer 1

Question :

The position function of a particle is x(t) = 30t −5t².

• Find the velocity function x'(t).
• Find the velocity when t = 2.
• Find the velocity when t = 4.
• When is the particle stationary?

Answer :

• Velocity function of the particle, x'(t) = (30 - 5t) m/s.
• The velocity of the particle at, t = 2, v₂ = 20 m/s.
• The velocity of the particle at, t = 4, v₄ = 10 m/s.
• The instànt at which the particle is at rest .i.e, v = 0, t = 6 s.

Explanation :

Given :

• Position function of the body, x(t) = 30t - 5t²

To find :

• Velocity function of the particle, x'(t) = ?
• The velocity of the particle at, t = 2, v₂ = ?
• The velocity of the particle at, t = 4, v₄ = ?
• The instànt at which the particle is at rest .i.e, v = 0, t = ?

Knowledge required :

• We know that the velocity of the particle is the derivative of the position of the particle w.r.t t .i.e, v = dx/dt

[Where : v = velocity of the particle, x = position of the particle.]

Let us take the unit of the velocity in S.I. system.i.e, m/s.

Solution :

To find the velocity function of the particle, x'(t) :

⠀⠀⠀By Using the equation for velocity of a particle, we get :

⠀⠀⠀=> v = d(30t - 5t²)/dt

⠀⠀⠀=> v = d(30t)dt - d(5t²)/dt

⠀⠀⠀=> v = 30 - 5t

∴ Hence the velocity function of the particle is (30 - 5t) m/s.

To find the velocity of the particle at, t = 2 :

⠀⠀⠀By Using the velocity function of the particle, we get :

⠀⠀⠀=> v₍ₜ ₌ ₂₎ = 30 - 5(2)

⠀⠀⠀=> v₍ₜ ₌ ₂₎ = 30 - 10

⠀⠀⠀=> v₍ₜ ₌ ₂₎ = 20

∴ Hence the velocity of the particle at,t = 2 s, is 20 m/s.

To find the velocity of the particle at, t = 4 :

⠀⠀⠀By Using the velocity function of the particle, we get :

⠀⠀⠀=> v₍ₜ ₌ ₄₎ = 30 - 5(4)

⠀⠀⠀=> v₍ₜ ₌ ₄₎ = 30 - 20

⠀⠀⠀=> v₍ₜ ₌ ₄₎ = 20

∴ Hence the velocity of the particle at,t = 4 s, is 10 m/s.

To find the instànt at which the particle is at rest .i.e, v = 0 :

According to the question, v = 0, thus,

⠀⠀⠀By Using the velocity function of the particle, we get :

⠀⠀⠀=> 0 = 30 - 5t

⠀⠀⠀=> -30 = -5t

⠀⠀⠀=> -30/-5 = t

⠀⠀⠀=> t = 6 s

∴ Hence at t = 6 s, the particle is at rest or stationary.