Question

An athlete starts at

point
A and runs at a constant
speed of around a circular track 100 m in diameter, as
shown in Fig. P3.40. Find the xand y-components of this runner’s average velocity and average acceleration between points
(a) A and B, (b) A and C, (c) C
and D, and (d) A and A (a full
lap). (e) Calculate the magnitude
of the runner’s average velocity
between A and B. Is his average speed equal to the magnitude of
his average velocity? Why or why not? (f) How can his velocity be
changing if he is running at constant speed?

Answer 1

Average velocity varies depending on displacement, while average acceleration is zero due to constant speed. Average speed is equal to average velocity magnitude for a full lap. Velocity changes direction despite constant speed due to circular path.

Average Velocity and Average Acceleration

Problem: An athlete starts at point A and runs at a constant speed of around a circular track 100 m in diameter. Find the x and y-components of the average velocity and average acceleration between points:

(a) A and B

(b) A and C

(c) C and D

(d) A and A (a full lap)

1. Define the variables:

Radius (r): 50 m (diameter is 100 m)

Constant speed (v): Unknown

Average velocity: (Vx, Vy)

Average acceleration: (Ax, Ay)

Time interval: Δt

2. Average Velocity:

The average velocity is given by the displacement divided by the time interval.

a) A and B:

Displacement: Δx = 50 m (horizontal)

Time interval: Δt = distance/speed = πr/v

Vx = Δx/Δt = 50v/πr

Vy = 0 (no vertical displacement)

b) A and C:

Displacement: Δx = -50 m (horizontal)

**Δy = 50 m (vertical)

Vx = -50v/πr

Vy = 50v/πr

c) C and D:

Displacement: Δx = -50 m (horizontal)

**Δy = -50 m (vertical)

Vx = -50v/πr

Vy = -50v/πr

d) A and A (full lap):

Displacement: Δx = 0

Δy = 0

Vx = 0

Vy = 0

3. Average Acceleration:

The average acceleration is given by the change in velocity divided by the time interval.

Since the athlete is running at a constant speed, the acceleration is zero.

Ax = Ay = 0 for all points.

4. Magnitude of Average Velocity:

The magnitude of the average velocity is given by:

|Vavg| = sqrt(Vx^2 + Vy^2)

a) A and B:

|Vavg(A,B)| = |50v/πr|

e) Comparison of Average Speed and Magnitude of Average Velocity:

Average speed is the total distance traveled divided by the total time taken. For a full lap, the average speed is equal to the magnitude of the average velocity because the athlete returns to the starting point.

f) Change in Velocity:

The athlete's velocity is changing in direction even though the speed is constant. This is because the athlete is moving in a circular path and the direction of the velocity vector is constantly changing.