Magnitude of average velocity and derive equation

[goldenroll]

These 2 questions are confusing me, I have some idea about what to do, can someone help?

Question 1

Homework Statement

Consider a circular track with a radius of 200 meters, suppose it takes 2 minutes to run from her starting point to the opposite side of the track. What is the magnitude of her average velocity over this 2 minute interval? What is her average speed over the same interval?

The Attempt at a Solution

I did 400m/120s and got 3.33 m/s, and for speed 400m/120s = 3.33 m/s (should I be putting significant figures also or something?)


Question 2

Homework Statement

Derive Vf^2=Vi^2+2ad

Homework Equations

(I think this is equation for velocity of an object)

The Attempt at a Solution

I thought that you would treat it like a derivative and comes out to 2Vi+2, but I got it wrong.

 

[rl.bhat]

Find the length of the track along the circular path.Now find her velocity. This will be her average speed.
Initially her velocity in one direction. When she reaches the diaametrically opposite point, what is the direction of her velocity and what is the magnitude of her velocity?
Now find the average velocity.

 

[tomcue]

The magnitude of average velocity is the absolute value of the average velocity, which is the distance traveled divided by the time taken. In this case, the magnitude of average velocity would be 3.33 m/s, as you have correctly calculated. For average speed, it is also the distance traveled divided by the time taken, but it does not take into account the direction of motion. Therefore, the average speed in this case would also be 3.33 m/s.

To derive the equation Vf^2=Vi^2+2ad, we can start with the definition of average velocity: Vavg = (Vf + Vi)/2. Since we are dealing with constant acceleration (a), we can use the equation Vf = Vi + at, where t is the time taken. Substituting this into the equation for average velocity, we get:

Vavg = (Vi + Vi + at)/2
Vavg = (2Vi + at)/2
Vavg = Vi + (at)/2

Now we can use the definition of acceleration (a = (Vf - Vi)/t) to substitute for (at)/2:

Vavg = Vi + (Vf - Vi)/2t
Vavg = (Vi + Vf)/2

We can also use the equation for displacement (d = Vit + 1/2at^2) and rearrange it to solve for Vf^2:

d = Vit + 1/2at^2
2d = 2Vit + at^2
2d = 2Vit + a(Vf - Vi)t
2d = 2Vit + aVft - aVit
2d = aVft + Vit
2d - Vit = aVft
2d/Vt - Vi = Vf

Substituting this into the equation for average velocity, we get:

Vavg = (Vi + Vf)/2
Vavg = (Vi + 2d/Vt - Vi)/2
Vavg = 2d/2Vt
Vavg = d/Vt

Solving for Vf^2, we get:

Vf^2 = (VavgVt)^2
Vf^2 = (d/Vt)^2

Substituting this into the equation 2d/Vt - Vi = Vf, we get:

Vf^2 =