Physics Practice Problems Motion in 2D & 3D With Calc Practice Problems Solution: An electron’s position is given by r = 3.00t î − ...

Solution: An electron’s position is given by r = 3.00t î − 4.00t2 ĵ + 2.00 k̂, with t in seconds and r in meters. (a) In unit-vector notation, what is the electron’s velocity v(t)? At t = 2.00 s, what is v (b) in unit-vector notation and as (c) a magnitude and (d) an angle relative to the positive direction of the x-axis?

Problem

An electron’s position is given by r = 3.00 − 4.00t2 + 2.00 , with t in seconds and r in meters. (a) In unit-vector notation, what is the electron’s velocity v(t)? At t = 2.00 s, what is v (b) in unit-vector notation and as (c) a magnitude and (d) an angle relative to the positive direction of the x-axis?

Solution

This problem requires us to determine a velocity in unit-vector notation, its magnitude, and its direction, given an object's position as a 3D function of time.

Anytime we're given a position, velocity, or acceleration function and asked to find one or more of the others, we know it's a motion problem with calculus. A PVA diagram like the one below can help remind you of the relationships between the three functions:

PVddtdtA

To get from position to velocity, we differentiate the position function. 

v(t)=dr(t)dt

Remember the power rule of differentiation. 

ddt(tn)=ntn-1  (for example, ddtt3=3t2)

Whenever you take the derivative of a vector, make sure to do the operation on each component (î, ĵ, and k̂) separately—they're independent of each other and shouldn't get mixed up!

Also recall the equations to find a vector's magnitude and direction from its components:

|v|=vx2+vy2

tanθ=vyvx

(a) We're asked to determine velocity function of the electron in unit-vector notation. 

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