CIRCULAR MOTION
A car takes a curve with 300m radius at a velocity of 60 m/s.
a) Determine the change in velocity (module and direction) when it travels an arc of 60 degrees
b)Compare the magnitude of the instantaneous acceleration of the car with a magnitude of the average acceleration to transverse the arc
Change in velocity
Δv⃗=v⃗₂-v⃗₁
|Δv|=sqrt(v²+v²-2v•v•cos60°)=sqrt(2v²(1-cos60°))=
=sqrt{2•60(1-0.5)}=7.74 m/s
The distance covered by the car is
s= 2•π•R•60°/360°=2•π•300/6=314 m
t= s/v=314/60=5.23 s
a=|Δv| /t=7.74/5.23 =1.48 m/s²
Centripetal acceleration (instantaneous acceleration)
a= v²/R= 60²/300=12 m/s²
Change in velocity
vecorΔv =vecor v₂-vecorv₁
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a) To determine the change in velocity (module and direction) as the car travels an arc of 60 degrees, we can use the concept of centripetal acceleration.
The formula for centripetal acceleration is:
ac = (v^2) / r
where ac is the centripetal acceleration, v is the velocity, and r is the radius of the curve.
Given the velocity of the car, v = 60 m/s, and the radius of the curve, r = 300 m, we can calculate the centripetal acceleration:
ac = (60^2) / 300
ac = 3600 / 300
ac = 12 m/s^2
Since the car is traveling along a curve, it experiences a change in direction. The velocity vector at any point on the curve is tangent to the curve. As the car moves along the 60-degree arc, the change in velocity is the change in magnitude and direction of this velocity vector.
To find the change in velocity, we need to determine the difference between the initial velocity vector and the final velocity vector after traveling the 60-degree arc.
The magnitude of the change in velocity can be calculated using the concept of angular velocity. The formula for angular velocity is:
ω = v / r
where ω is the angular velocity, v is the velocity, and r is the radius of the curve.
Given the velocity and radius, we have:
ω = 60 / 300
ω = 0.2 rad/s
To find the change in velocity, we multiply the angular velocity by the radius of the curve and the angle of the arc in radians:
Δv = ω * r * Δθ
where Δv is the change in velocity, ω is the angular velocity, r is the radius of the curve, and Δθ is the angle of the arc in radians.
Since the angle given is in degrees, we need to convert it to radians:
Δθ = (60 * π) / 180
Δθ = π / 3 radians
Plugging in the values, we have:
Δv = 0.2 * 300 * (π / 3)
Δv ≈ 62.83 m/s
Therefore, the change in velocity (module) when the car travels an arc of 60 degrees is approximately 62.83 m/s.
To determine the direction of the change in velocity, we need to consider the direction of the motion along the circular path. Since the car is traveling in a counterclockwise direction, the direction of the change in velocity will also be counterclockwise.
b) The magnitude of the instantaneous acceleration of the car can be found using the centripetal acceleration formula:
ac = (v^2) / r
Using the given values, we have:
ac = (60^2) / 300
ac = 3600 / 300
ac = 12 m/s^2
The magnitude of the average acceleration to traverse the arc can be found using the formula:
average acceleration = (change in velocity) / (time taken)
However, the time taken to travel the arc is not given. We can calculate the time taken by using the formula:
time = (arc length) / (velocity)
The arc length can be calculated using the formula:
arc length = (angle in radians) * (radius)
Given that the radius is 300m and the angle is 60 degrees, we convert the angle to radians:
angle in radians = (60 * π) / 180
angle in radians = π / 3
Plugging in the values, we have:
arc length = (π / 3) * 300
arc length ≈ 100π m
Now, let's calculate the time:
time = (100π) / 60
time ≈ 5.24 s
Now that we have calculated the time taken, we can find the average acceleration:
average acceleration = (change in velocity) / (time taken)
average acceleration = 62.83 / 5.24
average acceleration ≈ 11.99 m/s^2
Therefore, the magnitude of the instantaneous acceleration of the car is 12 m/s^2, and the magnitude of the average acceleration to traverse the arc is approximately 11.99 m/s^2.