Vector algebra - The velocity is given by

[lilyE]

Homework Statement

The velocity of an object as a function of time is given by:

Vx(t) = 12t² - 5t + 40 m/s
Vy(t) = 5t - 30 m/s

What is position at 2 sec if the object has an initial position of x = 5 m and y = 8 m?
What is the instantaneous acceleration at 10 s?

Homework Equations

The Attempt at a Solution

I started by just plugging time (2 sec) into the equation to get the velocity and then writing that out in component form (V = Vxi + Vyj = 78i - 20j), but it asks for position/acceleration and I'm not entirely sure how to go about solving for that.

 

[Fredrik]

Take the derivative to get the acceleration. Integrate to get the position.

 

[Doc Al]

How are position, velocity, and acceleration related? Hint: Calculus!

 

[lilyE]

Thank you! I think I've got it now.

 

[HalfLostAndSearching]

I would approach this problem by first understanding the concept of vector algebra and how it relates to the given equations. Vector algebra deals with the mathematical operations and properties of vectors, which are quantities that have both magnitude and direction.

In this case, the velocity of the object is given by two components, Vx and Vy, which represent the object's velocity in the x and y directions, respectively. The equations show that the velocity is a function of time, with the units of meters per second (m/s).

To find the position of the object at 2 seconds, we can use the position-velocity relationship, which states that the change in position (Δx or Δy) is equal to the velocity (Vx or Vy) multiplied by the change in time (Δt). So, for the x-direction, the position at 2 seconds would be:

Δx = Vx(2 sec) = (12*2^2 - 5*2 + 40) m = 78 m

Similarly, the position in the y-direction would be:

Δy = Vy(2 sec) = (5*2 - 30) m = -20 m

Therefore, the position of the object at 2 seconds would be (78, -20) m, with respect to its initial position of (5, 8) m.

To find the instantaneous acceleration at 10 seconds, we can use the definition of acceleration, which is the rate of change of velocity with respect to time. In other words, it is the derivative of the velocity function. So, for the x-direction, the acceleration at 10 seconds would be:

ax = dVx/dt = 24t - 5 m/s^2
at 10 seconds: ax(10 sec) = 24*10 - 5 m/s^2 = 235 m/s^2

Similarly, the acceleration in the y-direction would be:

ay = dVy/dt = 5 m/s^2
at 10 seconds: ay(10 sec) = 5 m/s^2

Therefore, the instantaneous acceleration at 10 seconds would be (235, 5) m/s^2.

In summary, vector algebra allows us to analyze the motion of an object in terms of its position, velocity, and acceleration in different directions. By understanding the equations and applying the appropriate mathematical operations, we can determine