A Particle, Initially At Rest, Moves Along The X X X -axis So That Its Acceleration At Any Time Is Given By A ( T ) = 8 T 2 − 6 A(t) = 8t^2 - 6 A ( T ) = 8 T 2 − 6 . Which Of The Following Is The Total Distance Traveled By The Particle Within The Time Interval $2 \leq

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Introduction

In physics, understanding the motion of particles is crucial for various applications, including engineering, mechanics, and even space exploration. One of the fundamental concepts in particle motion is acceleration, which is the rate of change of velocity. In this article, we will explore how to calculate the total distance traveled by a particle given its acceleration function.

Acceleration Function

The acceleration function of the particle is given by a(t)=8t2−6a(t) = 8t^2 - 6, where tt represents time. This function describes how the acceleration of the particle changes over time.

Velocity and Position Functions

To find the total distance traveled by the particle, we need to first find its velocity and position functions. The velocity function is the antiderivative of the acceleration function, and the position function is the antiderivative of the velocity function.

Velocity Function

The velocity function v(t)v(t) is the antiderivative of the acceleration function a(t)=8t2−6a(t) = 8t^2 - 6. We can find the velocity function by integrating the acceleration function with respect to time:

v(t)=∫a(t)dt=∫(8t2−6)dtv(t) = \int a(t) dt = \int (8t^2 - 6) dt

v(t)=8t33−6t+Cv(t) = \frac{8t^3}{3} - 6t + C

where CC is the constant of integration.

Position Function

The position function s(t)s(t) is the antiderivative of the velocity function v(t)v(t). We can find the position function by integrating the velocity function with respect to time:

s(t)=∫v(t)dt=∫(8t33−6t+C)dts(t) = \int v(t) dt = \int \left(\frac{8t^3}{3} - 6t + C\right) dt

s(t)=2t43−3t2+Ct+Ds(t) = \frac{2t^4}{3} - 3t^2 + Ct + D

where DD is the constant of integration.

Initial Conditions

The particle is initially at rest, which means that its initial velocity is zero. We can use this information to find the value of the constant of integration CC.

v(0)=0=8(0)33−6(0)+Cv(0) = 0 = \frac{8(0)^3}{3} - 6(0) + C

C=0C = 0

The initial position of the particle is not given, so we will leave the constant of integration DD as is.

Total Distance Traveled

The total distance traveled by the particle within the time interval 2≤t≤42 \leq t \leq 4 is given by the absolute value of the difference between the position at t=4t = 4 and the position at t=2t = 2:

D=∣s(4)−s(2)∣D = |s(4) - s(2)|

We can find the position at t=4t = 4 and t=2t = 2 by plugging these values into the position function:

s(4)=2(4)43−3(4)2+0s(4) = \frac{2(4)^4}{3} - 3(4)^2 + 0

s(4)=2(256)3−48s(4) = \frac{2(256)}{3} - 48

s(4)=5123−48s(4) = \frac{512}{3} - 48

s(4)=512−1443s(4) = \frac{512 - 144}{3}

s(4)=3683s(4) = \frac{368}{3}

s(4)=122.67s(4) = 122.67

s(2)=2(2)43−3(2)2+0s(2) = \frac{2(2)^4}{3} - 3(2)^2 + 0

s(2)=2(16)3−12s(2) = \frac{2(16)}{3} - 12

s(2)=323−12s(2) = \frac{32}{3} - 12

s(2)=32−363s(2) = \frac{32 - 36}{3}

s(2)=−43s(2) = \frac{-4}{3}

s(2)=−1.33s(2) = -1.33

Now we can find the total distance traveled by the particle:

D=∣s(4)−s(2)∣D = |s(4) - s(2)|

D=∣122.67−(−1.33)∣D = |122.67 - (-1.33)|

D=∣122.67+1.33∣D = |122.67 + 1.33|

D=∣124∣D = |124|

D=124D = 124

The total distance traveled by the particle within the time interval 2≤t≤42 \leq t \leq 4 is 124 units.

Conclusion

In this article, we have shown how to calculate the total distance traveled by a particle given its acceleration function. We have used the acceleration function a(t)=8t2−6a(t) = 8t^2 - 6 to find the velocity and position functions, and then used these functions to find the total distance traveled by the particle within the time interval 2≤t≤42 \leq t \leq 4. The total distance traveled by the particle is 124 units.

References

  • [1] Halliday, D., Resnick, R., & Walker, J. (2013). Fundamentals of Physics. John Wiley & Sons.
  • [2] Serway, R. A., & Jewett, J. W. (2018). Physics for Scientists and Engineers. Cengage Learning.

Glossary

  • Acceleration: The rate of change of velocity.
  • Velocity: The rate of change of position.
  • Position: The location of an object in space.
  • Distance: The total length of the path traveled by an object.

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Introduction

In our previous article, we explored how to calculate the total distance traveled by a particle given its acceleration function. In this article, we will answer some common questions related to this topic.

Q: What is the difference between distance and displacement?

A: Distance is the total length of the path traveled by an object, while displacement is the shortest distance between the initial and final positions of the object.

Q: How do I know if the particle is moving in the positive or negative direction?

A: To determine the direction of the particle's motion, you need to examine the sign of the velocity function. If the velocity function is positive, the particle is moving in the positive direction. If the velocity function is negative, the particle is moving in the negative direction.

Q: Can I use the acceleration function to find the total distance traveled by the particle?

A: No, you cannot use the acceleration function to find the total distance traveled by the particle. You need to first find the velocity function and then the position function to calculate the total distance traveled.

Q: What if the particle's acceleration function is not given? Can I still find the total distance traveled?

A: Yes, you can still find the total distance traveled by the particle even if the acceleration function is not given. You can use the velocity function and position function to calculate the total distance traveled.

Q: How do I know if the particle's motion is uniform or non-uniform?

A: To determine if the particle's motion is uniform or non-uniform, you need to examine the acceleration function. If the acceleration function is constant, the particle's motion is uniform. If the acceleration function is not constant, the particle's motion is non-uniform.

Q: Can I use the total distance traveled to find the particle's acceleration function?

A: No, you cannot use the total distance traveled to find the particle's acceleration function. You need to first find the acceleration function to calculate the total distance traveled.

Q: What if the particle's initial velocity is not given? Can I still find the total distance traveled?

A: Yes, you can still find the total distance traveled by the particle even if the initial velocity is not given. You can use the acceleration function and position function to calculate the total distance traveled.

Q: How do I know if the particle's motion is periodic or non-periodic?

A: To determine if the particle's motion is periodic or non-periodic, you need to examine the acceleration function. If the acceleration function is periodic, the particle's motion is periodic. If the acceleration function is not periodic, the particle's motion is non-periodic.

Conclusion

In this article, we have answered some common questions related to calculating the total distance traveled by a particle given its acceleration function. We hope that this article has provided you with a better understanding of this topic.

References

  • [1] Halliday, D., Resnick, R., & Walker, J. (2013). Fundamentals of Physics. John Wiley & Sons.
  • [2] Serway, R. A., & Jewett, J. W. (2018). Physics for Scientists and Engineers. Cengage Learning.

Glossary

  • Acceleration: The rate of change of velocity.
  • Velocity: The rate of change of position.
  • Position: The location of an object in space.
  • Distance: The total length of the path traveled by an object.
  • Displacement: The shortest distance between the initial and final positions of an object.
  • Periodic motion: Motion that repeats itself over a fixed time interval.
  • Non-periodic motion: Motion that does not repeat itself over a fixed time interval.