Position Vector Help: Solve for t = 9.00 s

[snoggerT]

The acceleration of a particle on a horizontal xy plane is given by a = 4t i + 5t j, where a is in meters per second-squared and t is in seconds. At t = 0, the particle has the position vector r = (20.0 m) i + (40.0 m) j and the velocity vector v = (5.00 m/s) i + (2.00 m/s) j.

(a) What is the position vector of the particle at t = 9.00 s?
r = ( m) i + ( m) j




This isn't a homework problem, but a example problem I'm trying to figure out for my test study guide. I've tried several ways, but can't seem to figure out the correct way at all. Any help to get me started would be great. Thanks.

 

[chaoseverlasting]

Ok. To start you off, here is what you need to use: $$a=\frac{d\vec{v}}{dt}$$ and $$v=\frac{d\vec{r}}{dt}$$.

 

[m2003]

Sure, I'd be happy to help you with this problem. First, let's review the given information:

- The particle's acceleration is given by a = 4t i + 5t j, where t is in seconds.
- At t = 0, the particle has a position vector of r = (20.0 m) i + (40.0 m) j.
- At t = 0, the particle has a velocity vector of v = (5.00 m/s) i + (2.00 m/s) j.

To solve for the position vector at t = 9.00 s, we will need to use the equations of motion. These are:

- v = u + at, where v is the final velocity, u is the initial velocity, a is the acceleration, and t is the time.
- r = ut + (1/2)at^2, where r is the final position, u is the initial velocity, a is the acceleration, and t is the time.

Let's use the first equation to find the final velocity at t = 9.00 s. We know that u = (5.00 m/s) i + (2.00 m/s) j, a = 4t i + 5t j, and t = 9.00 s. Plugging these values in, we get:

v = (5.00 m/s) i + (2.00 m/s) j + (4(9.00) m/s^2) i + (5(9.00) m/s^2) j

Simplifying, we get:

v = (41.00 m/s) i + (47.00 m/s) j

Now, let's use the second equation to find the final position at t = 9.00 s. We know that u = (5.00 m/s) i + (2.00 m/s) j, a = 4t i + 5t j, and t = 9.00 s. Plugging these values in, we get:

r = (5.00 m/s) i + (2.00 m/s) j + (1/2)(4(9.00)^2 m) i + (1/2)(5(9.00)^2 m) j

Simplifying, we get:

r = (5.00