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Displacement x=t^(3)-12t+10. The acceler...

Displacement `x=t^(3)-12t+10`. The acceleration of particle when velocity is zero is given by

A

12 units

B

10 units

C

2 units

D

1 units

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To solve the problem, we need to find the acceleration of the particle when its velocity is zero. Here’s a step-by-step solution: ### Step 1: Write down the displacement equation The displacement of the particle is given by: \[ x = t^3 - 12t + 10 \] ### Step 2: Find the velocity The velocity \( v \) is the first derivative of displacement with respect to time \( t \): \[ v = \frac{dx}{dt} = \frac{d}{dt}(t^3 - 12t + 10) \] Calculating the derivative: \[ v = 3t^2 - 12 \] ### Step 3: Set the velocity to zero and solve for \( t \) To find when the velocity is zero, we set the equation equal to zero: \[ 3t^2 - 12 = 0 \] Solving for \( t \): \[ 3t^2 = 12 \] \[ t^2 = 4 \] \[ t = 2 \quad \text{(taking the positive root since time cannot be negative)} \] ### Step 4: Find the acceleration The acceleration \( a \) is the second derivative of displacement with respect to time, or the first derivative of velocity: \[ a = \frac{d^2x}{dt^2} = \frac{dv}{dt} = \frac{d}{dt}(3t^2 - 12) \] Calculating the derivative: \[ a = 6t \] ### Step 5: Substitute \( t = 2 \) into the acceleration equation Now, we substitute \( t = 2 \) into the acceleration equation: \[ a = 6(2) = 12 \] ### Conclusion The acceleration of the particle when the velocity is zero is: \[ a = 12 \, \text{units} \] ---
To solve the problem, we need to find the acceleration of the particle when its velocity is zero. Here’s a step-by-step solution: ### Step 1: Write down the displacement equation The displacement of the particle is given by: \[ x = t^3 - 12t + 10 \] ### Step 2: Find the velocity The velocity \( v \) is the first derivative of displacement with respect to time \( t \): ...
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