Question

A particle is moving in a straight line. The variation of position \( x \) as a function of time \( t \) is given as\[x = (t^3 - 6t^2 + 20t + 15) \, \text{m}.\]The velocity of the body when its acceleration becomes zero is:

• A. 4 m/s
• B. 8 m/s
• C. 10 m/s
• D. 6 m/s

Answer 1

The Correct Option is B

To determine the velocity of the particle when its acceleration becomes zero, we first need to understand the relationship between position, velocity, and acceleration for this motion. 

1. Given the position of the particle as a function of time: \(x = t^3 - 6t^2 + 20t + 15 \, \text{m}\)
2. The velocity \((v)\) is the first derivative of the position with respect to time:

\(v = \frac{dx}{dt} = \frac{d}{dt}(t^3 - 6t^2 + 20t + 15)\)

Calculating the derivative, we get: \(v = 3t^2 - 12t + 20\)

1. The acceleration \((a)\) is the first derivative of velocity (or the second derivative of position) with respect to time:

\(a = \frac{dv}{dt} = \frac{d}{dt}(3t^2 - 12t + 20)\)

Calculating the derivative, we get: \(a = 6t - 12\)

1. We need the acceleration to be zero to find the corresponding time:

\(6t - 12 = 0\)

Solving for \(t\), we get: \(6t = 12 \quad \Rightarrow \quad t = 2\)

1. Substitute \(t = 2\) into the velocity expression to find the velocity when the acceleration is zero:

\(v = 3(2)^2 - 12(2) + 20\)

\(v = 3(4) - 24 + 20\)

\(v = 12 - 24 + 20\)

\(v = 8 \, \text{m/s}\)

1. Therefore, the velocity of the body when its acceleration becomes zero is 8 m/s.

Hence, the correct answer is 8 m/s.

Answer 2

Given the position function:  
\(x = t^3 - 6t^2 + 20t + 15\)

Step 1: Find the velocity \( v \):  
The velocity is the first derivative of the position function with respect to time:  
\(v = \frac{dx}{dt} = 3t^2 - 12t + 20\)

Step 2: Find the acceleration \( a \):  
The acceleration is the derivative of velocity:  
\(a = \frac{dv}{dt} = 6t - 12\)

Step 3: When is the acceleration zero?  
Set the acceleration to zero to find the time at which the acceleration becomes zero:  
\(6t - 12 = 0 \implies t = 2 \, \text{sec}\)

Step 4: Find the velocity at \( t = 2 \):  
Substitute \( t = 2 \) into the velocity equation:  
\(v = 3(2)^2 - 12(2) + 20 = 3(4) - 24 + 20 = 12 - 24 + 20 = 8 \, \text{m/s}\)

Thus, the velocity of the body when its acceleration becomes zero is.  \( 8 \, \text{m/s} \)

The Correct Answer is:  \( 8 \, \text{m/s} \)