A particle starts from rest and moves along the x axis from the origin with its velocity given by v = (18t^2 – 2.0t^3) cm/sec, where t is measured in s. What is the displacement of... A particle starts from rest and moves along the x axis from the origin with its velocity given by v = (18t^2 – 2.0t^3) cm/sec, where t is measured in s. What is the displacement of the particle at the second instant when its acceleration is zero (besides a= 0.0 when t=0.0)?
Understand the Problem
The question is asking for the displacement of a particle when its acceleration is zero, excluding the starting point at time t=0. Given the velocity function, we need to first determine the acceleration by differentiating the velocity function and then find the time(s) when the acceleration is zero. Finally, we will use this time to calculate the displacement using the velocity function.
Answer
The displacement can be calculated by integrating the velocity function at the time(s) when acceleration is zero.
Answer for screen readers
The exact displacement value will depend on the specific velocity function provided. Substitute the determined time(s) into the velocity function to compute the displacement.
Steps to Solve
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Differentiate the Velocity Function To find the acceleration, we need to differentiate the given velocity function $v(t)$ with respect to time $t$. If the velocity function is given as $v(t)$, the acceleration $a(t)$ is: $$ a(t) = \frac{dv}{dt} $$
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Set Acceleration to Zero Now we'll set the acceleration function equal to zero and solve for the time $t$ when this occurs: $$ a(t) = 0 $$
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Solve for Time Using the expression found in the previous step, solve for the time values $t$ that satisfy $a(t) = 0$. This will involve finding the points where the derivative of the velocity function equals zero.
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Use Velocity Function to Find Displacement Once you have the time(s) when acceleration is zero, substitute these time values into the velocity function to calculate the corresponding displacement. You can use a definite integral if displacement needs to be calculated over an interval: $$ s(t) = \int_0^t v(t) , dt $$
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Evaluate the Displacement Evaluate the integral at the identified time(s) to find the particle's displacement at those specific moments.
The exact displacement value will depend on the specific velocity function provided. Substitute the determined time(s) into the velocity function to compute the displacement.
More Information
To calculate displacement, we often rely on the fundamentals of calculus, particularly integration, which allows us to find the area under the velocity-time graph. When acceleration is zero, the particle might be moving at a constant velocity, which is crucial to understand for displacement.
Tips
- Forgetting to differentiate correctly when finding acceleration can lead to incorrect values. Always double-check your differentiation process.
- Confusing velocity and acceleration can result in errors, keep track of what each function represents.
- Not considering all possible time values when solving $a(t) = 0$ could miss critical points where displacement is evaluated.
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