A car travels around a bend of radius R at a speed v1, experiencing a centripetal acceleration ac. The car then travels around a curve with the same radius R, but with a centripetal acceleration which has twice the magnitude of ac. What speed is the car moving, when going around the second curve?
a. v1/(square root of 2)
b. square root of (2 times v1)
c. v1
d. square root of 2 , times v1
Centripetal acceleration
=v²/r
or it varies with the square of the velocity.
When acceleration is doubled, the velocity is increased by a factor of √2.
This way (√2)²=2.
To solve this problem, we need to use the concept of centripetal acceleration and the relationship between speed, radius, and centripetal acceleration.
Centripetal acceleration (ac) can be calculated using the formula: ac = v^2 / R
In the first scenario, the car travels around the bend with radius R at a speed v1, experiencing centripetal acceleration ac. Therefore, we have ac = v1^2 / R.
In the second scenario, the car travels around a curve with the same radius R but with a centripetal acceleration that has twice the magnitude of ac. Let's call this new centripetal acceleration ac2. Therefore, we have ac2 = 2 * ac, or ac2 = 2 * (v1^2 / R).
Now, let's find the speed of the car when going around the second curve. We know that ac2 = v^2 / R, where v is the speed of the car.
Substituting the values we have for ac2 and v2 into the equation, we get:
2 * (v1^2 / R) = v^2 / R
Simplifying the equation, we get:
2 * v1^2 = v^2
Taking the square root of both sides, we get:
sqrt(2 * v1^2) = sqrt(v^2)
Simplifying further, we have:
sqrt(2) * v1 = v
Therefore, the speed of the car when going around the second curve is equal to v1 multiplied by the square root of 2. Hence, the correct answer is option d) square root of 2, times v1.