I'm given a question to suppose that an ice skater on a frictionless surface is attempting to reduce his speed. His initial speed is $v_0$. His deceleration (negative acceleration) is purely due to air resistance, given by $F=-kmv^2$, where $m$ is the skater's mass and $k$ is a constant. From the time when he begins to decelerate, what is his speed at time $t$?

Obviously, firstly I recognise the force of deceleration $F=ma$ is such that $ma=-kmv^2$ Therefore, $$a=-kv^2$$

To find the speed at time $t$, it should be such that: $$\int a \ dt = \int -kv^2 \ dt $$ $$v=-kv^2t$$

But what I'm confused is that the acceleration/deceleration at a given time already depends on its speed which means my previous equation is already wrong since I cannot treat $v$ as a constant. How am I supposed to take this into consideration to find the speed?

Thank you!

I'm given a question to suppose that an ice skater on a frictionless surface is attempting to reduce his speed. His initial speed is $v_0$. His deceleration (negative acceleration) is purely due to air resistance, given by $F=-kmv^2$, where $m$ is the skater's mass and $k$ is a constant. From the time when he begins to decelerate, what is his speed at time $t$?

Obviously, firstly I recognise the force of deceleration $F=ma$ is such that $ma=-kmv^2$ Therefore, $$a=-kv^2$$

To find the speed at time $t$, it should be such that: $$\int a \ dt = \int -kv^2 \ dt $$ $$v=-kv^2t$$

But what I'm confused is that the acceleration/deceleration at a given time already depends on its speed which means my previous equation is already wrong since I cannot treat $v$ as a constant. How am I supposed to take this into consideration to find the speed?