Question

as a car speed up when rounding a curve does it centripetal acceleration increases ? use an equation to defend your answer ​

Answer 1

This question can be answered with only facts also. let us see how

We know, Centripetal acceleration = v² / r

Since the curve is round which means radius would be same while the car travel the curve.

Now, It is given in the car that the car speeds up. But centripetal acceleration is directly proportional to the velocity.

Since, Velocity is increasing here. Therefore, Centripetal acceleration would also increase.

Now, Let us do this by using equation.

Let at the start of the car velocity be v + x.

⇒ Centripetal a at that place = (v + x)² / r ...(1)

Now, Let us look after say ∆t passes and so the velocity would increase, say by y, Hence new velocity would be v + x + y

⇒ Centripetal a at that place = (v + x + y)² / r ...(2)

Further, Let the new velocity be v + x + y + z after ∆t₂ passes.

⇒ Centripetal at that place = (v + x + y + z)² / r ..(3)

To Be Noted : (v + x) < (v + x + y) < (v + x + y + z) ...(4)

Now, Let us find the ratio of centripetal acceleration:

⇒ (1) : (2) : (3) = (v + x)² / r : (v + x + y)² / r : (v + x + y + z)² / r

⇒ (1) : (2) : (3) = (v + x)² : (v + x + y)² : (v + x + y + z)²

But from (4),

⇒ (v + x) < (v + x + y) < (v + x + y + z)

∴ (1) < (2) < (3)

Which means Centripetal acceleration at (1) is less than centripetal acceleration at (2) which is less than centripetal acceleration at (3)

This indicates that the centripetal acceleration also increased as the velocity did.