using d^2(h/dt^2=−g+k(�dh/dt^2, find an expression for the terminal velocity in terms of k and g? g is the acceleration due to gravity, k is a constant, h(t) is the height of the falling object, dh/dt is its velocity, and d^2(h)dt^2 is its acceleration. And can you explain your answer as well thatd be great!

Question

using d^2(h/dt^2=−g+k(�dh/dt^2, find an expression for the terminal velocity in terms of k and g? g is the acceleration due to gravity, k is a constant, h(t) is the height of the falling object, dh/dt is its velocity, and d^2(h)dt^2 is its acceleration. And can you explain your answer as well thatd be great!

Answer 1

To find the expression for the terminal velocity in terms of k and g, we can start by considering the equation for the acceleration of a falling object:

d^2(h)/dt^2 = -g + k(dh/dt)^2

Here, d^2(h)/dt^2 represents the acceleration of the object, g is the acceleration due to gravity, k is a constant, and dh/dt represents its velocity.

Next, let's assume that the object has reached its terminal velocity. Terminal velocity occurs when the acceleration of the object becomes zero because the object is no longer accelerating, but is moving at a constant speed.

So, we can set d^2(h)/dt^2 = 0:

0 = -g + k(dh/dt)^2

Now, we can solve this equation for dh/dt, which represents the terminal velocity (v_t):

g = k(v_t)^2

To isolate v_t, we divide both sides of the equation by k:

v_t^2 = g/k

Finally, we take the square root of both sides to get the expression for the terminal velocity:

v_t = sqrt(g/k)

Therefore, the expression for the terminal velocity in terms of k and g is given by sqrt(g/k).

In summary, we started with the equation of motion for the falling object, assumed it had reached its terminal velocity, set the acceleration equal to zero, and then solved for the terminal velocity in terms of k and g.