the position of a particular particle as a function of time is given by r⃗ =(9.60t i + 8.85j + 1.00t^2 k) . determine tge particles average velocityand acceleration
average over what interval?
You know that
v = 9.6 i + 2.00t k
a = 2.00 k
Once you decide on your interval [a,b] for t, then the average
velocity is, as always, (v(b)-v(a))/(b-a).
The acceleration is, of course, constant.
To determine the particle's average velocity and acceleration, we need to use the given position vector as a function of time.
The position vector is given as r⃗ = (9.60t i + 8.85j + 1.00t^2 k), where i, j, and k are unit vectors in the x, y, and z directions respectively.
1. Average velocity:
The average velocity of the particle is given by the change in position divided by the change in time. To find the change in position, we subtract the initial position from the final position.
Let's assume the initial time is t1 and the final time is t2.
Change in position, Δr⃗ = r⃗(t2) - r⃗(t1)
= (9.60t2 i + 8.85j + 1.00t2^2 k) - (9.60t1 i + 8.85j + 1.00t1^2 k)
= (9.60(t2-t1) i + (t2^2-t1^2) k)
Change in time, Δt = t2 - t1
Average velocity, v⃗ average = Δr⃗ / Δt
= (9.60(t2-t1) i + (t2^2-t1^2) k) / (t2 - t1)
2. Average acceleration:
The average acceleration of the particle is given by the change in velocity divided by the change in time. To find the change in velocity, we subtract the initial velocity from the final velocity.
Change in velocity, Δv⃗ = v⃗(t2) - v⃗(t1)
Using the average velocity equation, v⃗ average = Δr⃗ / Δt, we can substitute it into the change in velocity equation.
So, Δv⃗ = (9.60(t2-t1) i + (t2^2-t1^2) k)
Average acceleration, a⃗ average = Δv⃗ / Δt
= (9.60(t2-t1) i + (t2^2-t1^2) k) / (t2 - t1)
These formulas will help you calculate the particle's average velocity and acceleration using the given position vector. Just substitute the appropriate values of t1 and t2 to get the numerical result.