determine the average acceleration of a slalom skier if she is moving at a velocity of 9.75 m/s[S25E] and then changes her velocity of 8.64 m/s [W35S] in a matter of 0.179sec.
original south = 9.75 cos 25
final south = 8.64 sin 35
original east = 9.75 sin 25
final east = -8.64 cos 35
south a component = (final south -original south)/ .179
east a component = (final east component - original east component) / .179
resultant mag of a = sqrt (south a^2 + east a^2 )
angle T south of east
tan T = final a south / final a east
acceleration=(vfinal-Vinitial)/time
Now, notice the velocity is vectors. I would do it like this
vf=9.75cos25 S + 9.75sin25 E
Vi=8.64sin35 S + 8.64cos35 W
remember W is -E, so when you subtract..
Vf-Vi=(9.75cos25 -8.64sin35)S +(9.75sin25+8.64cos35) E
do that math, divide by time
If you wish, change it to polar form, the problem did not require it.
To determine the average acceleration of a slalom skier, we'll use the formula:
Average Acceleration = (Final Velocity - Initial Velocity) / Time
Given:
Initial Velocity (v1) = 9.75 m/s [S25E]
Final Velocity (v2) = 8.64 m/s [W35S]
Time (t) = 0.179 sec
Step 1: Calculate the change in velocity
Δv = v2 - v1 = (8.64 m/s [W35S]) - (9.75 m/s [S25E])
To calculate the change in velocity, we need to subtract the initial velocity from the final velocity. However, we need to account for the direction of the velocities. The addition and subtraction of vectors require using vector addition rules, and it involves both magnitude and direction.
To simplify this process, we'll convert the velocities to their respective components. Let's start by converting the compass directions (S25E, W35S) into their corresponding positive/negative x and y components:
S25E can be divided into separate south and east components:
- The south component is -25 m/s (negative because it's in the opposite direction of the positive y-axis).
- The east component is +25 m/s (positive because it's in the positive x-axis direction).
W35S can be divided into separate west and south components:
- The west component is -35 m/s (negative because it's in the opposite direction of the positive x-axis).
- The south component is -35 m/s (negative because it's in the opposite direction of the positive y-axis).
Now let's calculate the x-component and y-component separately.
X-component:
v1_x = v1 * cos(θ)
v2_x = v2 * cos(θ)
For v1, θ = atan(South component / East component)
For v2, θ = atan(South component / West component)
Note: atan() denotes the inverse tangent function.
Y-component:
v1_y = v1 * sin(θ)
v2_y = v2 * sin(θ)
For v1, θ = atan(South component / East component)
For v2, θ = atan(South component / West component)
Using the given compass directions and the information above, calculate the x and y components of both the initial and final velocities.
Step 2: Calculate the change in velocity components
Δv_x = v2_x - v1_x
Δv_y = v2_y - v1_y
Substitute the calculated x and y components from Step 1 into the formula.
Step 3: Calculate the magnitude of the change in velocity vector
Δv = sqrt(Δv_x^2 + Δv_y^2)
Calculate the square root of the sum of the squares of the x and y components from Step 2 to find the magnitude of the change in velocity vector.
Step 4: Calculate the average acceleration
Average Acceleration = Δv / t
Substitute the calculated change in velocity (magnitude) and the provided time into the formula to find the average acceleration.