49. The displacement of a particle after time t is given by x = (t3 -6+2+31+4) metre. What is
the velocity of the particle when its acceleration is zero
1) - 12 m/s
2) - 9 m/s
3) - 6 m/s
4)-3 m/s
Answers
Question :
The displacement of a particle after time t is given by x = (t³ - 4t²+ 3t) metre. What is
the velocity of the particle,when its acceleration is zero.
Explanation :
Given :
- Displacement of the particle, x = (t³ - 4t² + 3t)
- Acceleration of the particle, a = 0 m/s²
To find :
- Velocity of the particle when the acceleration produced by the particle is 0, v = ?
Knowledge required :
- Differentiation of displacement of a particle gives the velocity of that particle.
So the formula for Velocity of a particle is the change of change of position of the particle with change time, i.e, v = d(x)/dt.
- Differentiation of acceleration of yhe particle gives the acceleration of that particle.
So the formula for acceleration of a particle is the change of change of velocity of the particle with change in time, i.e, a = d(v)/dt.
- Differentiation of a constant term is 0,i.e, d(c)/dx = 0, [Where, c is a constant]
- Power rule of differentiation,i.e d(xⁿ)/dx = x⁽ⁿ ⁻ ¹⁾.
Solution :
To find the velocity of the particle :
⠀By using the equation for velocity of a particle and differentiating it w.r.t time, we get :
⠀⠀=> v = d(x)/dt
⠀⠀=> v = d(t³ - 4t² + 3t)/dt
⠀⠀=> v = d(t³)/dt - d(4t²)/dt + d(3t)/dt + d(4)/dt
⠀⠀=> v = [3 × t⁽³ ⁻ ¹⁾] - [2 × 4t⁽² ⁻ ¹⁾] + [1 × 3t⁽¹ ⁻ ¹⁾]
⠀⠀=> v = [3 × t²] - [2 × 4t¹] + [1 × 3t⁰]
⠀⠀=> v = 3t² - 8t + 3
⠀⠀⠀∴ v = (3t² - 8t + 3) m/s
Hence the velocity of the particle is (3t² - 12t + 3) m/s.
To find the acceleration of the particle :
⠀By using the equation for acceleration of a particle and differentiating it w.r.t time, we get :
⠀⠀=> a = d(v)/dt
⠀⠀=> a = d(3t² - 8t + 3)/dt
⠀⠀=> a = d(3t²)/dt - d(8t)/dt + d(3)/dt
⠀⠀=> a = [2 × 3t⁽² ⁻ ¹⁾] - [1 × 8t⁽¹ ⁻ ¹⁾] + 0
⠀⠀=> a = [2 × 3t¹] - [1 × 8t⁰]
⠀⠀=> a = 6t - 8
⠀⠀⠀∴ a = (6t - 8) m/s²
Hence the acceleration of the particle is (6t - 8) m/s².
To find the instant of time when the acceleration produced by the particle is zero :
⠀By using the acceleration of the particle and substituting the values in it, we get :
⠀⠀=> a = 6t - 8
⠀⠀=> a = 6t - 8
⠀⠀=> 8 = 6t
⠀⠀=> 8/6 = t
⠀⠀=> 1.4 = t
⠀⠀⠀⠀∴ t = 1.4 s
Hence the instant of time when the acceleration produced by the particle is zero is 2 s.
To find the velocity of the particle at t = 2 s.
By using the velocity of the particle and finding it's velocity at, t = 2 s , we get :
⠀⠀=> v = 3t² - 12t + 3
⠀⠀=> v = 3(1.4)² - 12(1.4) + 3
⠀⠀=> v = 5.88 - 16.8 + 3
⠀⠀=> v = -7.92 or - 8(approx.)
⠀⠀⠀∴ v = -8 m/s
Hence,Velocity of the particle when the acceleration produced by the particle is 0, v = -8 m/s
- The displacement of a particle after time t is given by .
- The velocity of the particle when it's acceleration is zero.
☆ It is given that acceleration is zero.
☆ Now put the value of t in equation(1), we get
2) The velocity of the particle is -9m/s.