A force of 40N is responsible for the motion of a
body governed by the equation s = 2t + 2t2 where
s is in metres and t in sec. What is the momentum
of the body at t = 2 sec?
[Hint: Find acc. then m = F/a & p = mv]
8. A body whose mass is 3 kg performs rec
Answers
Answer:
Step-by-step explanation:
GIVEN :-
S = 2t + t^2
F = 40 N
t =2 SEC
P = ?
a = 4 M/S^2 .
as ,
from Newton's second law of motion .
F = M * A
M = F / A
= 40/4
=10 KG
THEREFORE :-
V = 2 + 4 * t
=2 + 4 *2
=10 m/s
as momentum = mass into acc^n
P = m * v
P = 10 * 10
P = 100 Kg m/s
Answer :
- Momentum of the body at the instant of 2 s is 100 kg m/s.
Explanation :
Given :
- Force responsible for the motion of the body, F = 40 N
- Displacement of the body, s = 2t + 2t²
- Instant of time, t = 2 s
To find :
- The momentum of the body, p = ?
Knowledge required :
- Differentiation of the displacement of a body gives the velocity of the body.
So the formula for velocity of a body will be, v = d(s)/dt.
- Differentiation of the velocity of a body gives the accelaration of the body.
So the formula for accelaration of a body will be, a = d(v)/dt.
- Derivative of a constant term is 0, i.e, d(c)/dx = 0.
- Power rule of differentiation, d(x^n)/dx = n·x^(n - 1).
- Formula for force exerted by a body, F = ma.
[Where : m = Mass of the body, a = Acceleration produced by the body, F = Force exerted by the body]
- Formula for momentum of a body, p = mv.
[Where : m = Mass of the body, v = Velocity the body, p = Momentum of the body]
Solution :
To find the velocity of the body :
⠀By using the formula for velocity of a body and differentiating with respect to t, we get :
⠀⠀=> v = d(s)/dt
⠀⠀=> v = d(s)/dt = d(2t + 2t²)/dt
⠀⠀=> v = d(s)/dt = d(2t)/dt + d(2t²)/dt
⠀⠀=> v = d(s)/dt = [1 × 2t⁽¹ ⁻ ¹⁾] + [2 × 2t⁽² ⁻ ¹⁾]
⠀⠀=> v = d(s)/dt = [1 × 2t⁰) + [2 × 2t¹]
⠀⠀=> v = d(s)/dt = 2 + 2 × 2t
⠀⠀=> v = d(s)/dt = 2 + 4t
⠀⠀⠀⠀⠀⠀∴ v = 2 + 4t
Hence,
- Velocity of the body, v = (2 + 4t) m/s
Now let's find out the velocity of the body at the instant of 2 s (i.e, t = 2 s).
⠀⠀=> v = 2 + 4t
⠀⠀=> v₍ₜ ₌ ₂ ₛ₎ = 2 + 4(2)
⠀⠀=> v₍ₜ ₌ ₂ ₛ₎ = 2 + 8
⠀⠀=> v₍ₜ ₌ ₂ ₛ₎ = 10
⠀⠀⠀⠀⠀⠀∴ v = 10 m/s
Hence,
- Velocity of the body at t = 2 s, v = 10 m/s.
To find the acceleration of the body :
⠀By using the formula for accelaration of a body and differentiating with respect to t, we get :
⠀⠀=> a = d(v)/dt
⠀⠀=> a = d(v)/dt = d(2 + 4t)/dt
⠀⠀=> a = d(v)/dt = d(2)/dt + d(4t)/dt
⠀⠀=> a = d(v)/dt = 0 + [1 × 4t⁽¹ ⁻ ¹⁾]
⠀⠀=> a = d(v)/dt = 4
⠀⠀⠀⠀⠀⠀∴ a = 4 m/s²
Hence,
- Acceleration of the body, v = 4 m/s².
Now let's find out the acceleration of the body at the instant of 2 s (i.e, t = 2 s).
From the above calculation, we can conclude that the acceleration of the body is constant at any instant of time.
Mathematically,
⠀⠀=> a = 4
⠀⠀=> a₍ₜ ₌ ₂ ₛ₎ = 4
⠀⠀⠀⠀⠀⠀∴ a = 4 m/s²
Hence,
- Acceleration of the body at t = 2 s, a = 4 m/s².
To find the mass of the body :
⠀By using the formula for force exerted by a body and substituting the values in it, we get :
⠀⠀=> F = ma
⠀⠀=> 40 = m(4)
⠀⠀=> 40/4 = m
⠀⠀=> 10 = m
⠀⠀⠀⠀∴ m = 10 kg
Hence,
- Mass of the body, m = 10 kg
To find the momentum of the body :
⠀By using the formula for momentum of a body and substituting the values in it, we get :
⠀⠀=> p = mv
⠀⠀=> p = 10 × 10
⠀⠀=> F = 10
⠀⠀⠀⠀∴ p = 10 kg m/s
Hence,
- Momentum of the body, p = 10 kg m/s