An automobile travels at a constant speed around a curve whose radius of curvature is 2000m. What is the maximum allowable speed if the maximum acceptable value for the normal scalar component of acceleration is 1.5m/s^2?
To find the maximum allowable speed for the automobile, we need to consider the relationship between the normal scalar component of acceleration and the speed around a curve.
The formula that relates these two variables is:
a_n = v^2 / r
Where:
a_n is the normal scalar component of acceleration,
v is the speed of the automobile, and
r is the radius of curvature.
In this case, we have:
r = 2000m
a_n = 1.5m/s^2
Substituting these values into the formula, we can solve for v:
1.5m/s^2 = v^2 / 2000m
To isolate v^2, we need to multiply both sides of the equation by 2000m:
1.5m/s^2 * 2000m = v^2
3000m^2/s^2 = v^2
Finally, we can take the square root of both sides of the equation to solve for v:
v = √(3000m^2/s^2)
v ≈ 54.77 m/s
Therefore, the maximum allowable speed for the automobile is approximately 54.77 m/s.
To find the maximum allowable speed, we can use the concept of centripetal acceleration. The centripetal acceleration of an object moving in a circle is given by the formula:
a = v^2 / r
Where:
a = centripetal acceleration
v = velocity of the object
r = radius of curvature
In this case, we want to find the maximum speed at which the normal scalar component of acceleration, which is the centripetal acceleration, does not exceed 1.5m/s^2.
We can rearrange the formula to solve for the velocity:
v = √(a * r)
Substituting the given values:
a = 1.5m/s^2
r = 2000m
v = √(1.5 * 2000)
v = √(3000)
v ≈ 54.77 m/s
Therefore, the maximum allowable speed is approximately 54.77 m/s.