What is the acceleration vector for the given position and velocity vectors?

In summary, acceleration is the rate of change of an object's velocity over time and is important for understanding an object's future motion and the forces acting upon it. To find the acceleration vector, one must measure the change in velocity over a specific time interval. There is a difference between average and instantaneous acceleration, with average giving an overall view of motion and instantaneous showing the change at a specific moment. The acceleration of an object can be affected by factors such as net force, mass, and direction, as well as external forces like friction and air resistance. Finally, acceleration is related to Newton's laws of motion, specifically the second and third laws.
  • #1
cue928
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For the following position vector I am being asked to find the acceleration at t=1:
r(t) = (t^5)i + 8ln(1/(6+t))j + (9/t)k

I had the following for the velocity vector:
v(t) = (5t^4)i + (-8/(6+t))j - (9/t^2)k

For the acceleration vector I had:
a(t) = (20t^3)i + (8/(t+6)^2)j + (18/t^3)k

a(1) = 20 i + (8/49)j + 18 k

According to the book, a(1) = 20i + 8j + 18k is the correct answer, does anyone see where I went wrong or is it a mistake in the book?
 
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  • #2
The book's wrong.
 

What is acceleration and why is it important?

Acceleration is the rate of change of an object's velocity over time. It is important because it helps us understand how fast an object's velocity is changing, which is crucial for predicting its future motion and determining the forces acting upon it.

How do you find the acceleration vector of an object?

To find the acceleration vector, you need to measure the change in an object's velocity over a specific time interval. This can be done by measuring the initial and final velocities and the time it takes for the change to occur. The acceleration vector will have both magnitude and direction, and can be represented as a vector arrow.

What is the difference between average and instantaneous acceleration?

Average acceleration is the change in velocity over a specific time interval, while instantaneous acceleration is the acceleration at a specific moment in time. Average acceleration gives an overall view of an object's motion, while instantaneous acceleration shows how the object's velocity is changing at a specific point.

What factors can affect the acceleration of an object?

The acceleration of an object can be affected by various factors, including the net force acting on the object, the mass of the object, and the direction of the force. Other factors such as friction and air resistance can also play a role in the acceleration of an object.

How is acceleration related to Newton's laws of motion?

According to Newton's second law of motion, the acceleration of an object is directly proportional to the net force acting on it and inversely proportional to its mass. This means that the greater the force acting on an object, the greater its acceleration will be. Additionally, Newton's third law states that for every action, there is an equal and opposite reaction, which can also impact the acceleration of an object.

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