A cart released from rest at the top of a track 90. cm long takes 4.00 seconds to reach the bottom. determine the velocity after half the time.

To determine the velocity of the cart after half the time, we need to calculate the time it takes for the cart to reach half the distance.

Given:
Length of the track (h): 90 cm
Total time taken (t): 4.00 seconds

First, let's find the time it takes to reach half the distance.

Half the distance is given by: (1/2) * h = (1/2) * 90 cm = 45 cm.

Now, we can calculate the time it takes to cover half the distance using the equation of motion:

s = ut + (1/2) * a * t^2

Here, s is the distance travelled, u is the initial velocity (which is zero since the cart is released from rest), a is the acceleration, and t is the time.

Since we are interested in the time it takes to reach half the distance, we can rewrite the equation as:

(1/2) * h = (1/2) * a * (t/2)^2

Substituting the values, we have:

45 cm = (1/2) * a * (t/2)^2

Now, we can solve for the value of a using the given total time of 4.00 seconds:

45 cm = (1/2) * a * (4.00/2)^2

45 cm = (1/2) * a * 1^2
45 cm = (1/2) * a

Multiplying both sides by 2, we have:

90 cm = a

Now we have the acceleration. Next, we can find the time it takes to reach half the distance (45 cm) using the equation of motion:

s = ut + (1/2) * a * t^2

45 cm = 0 + (1/2) * 90 cm * t^2

Simplifying, we have:

t^2 = 1
t = 1 second

Therefore, it takes 1 second for the cart to reach half the distance.

Now, let's calculate the velocity after half the time using the equation of motion:

v = u + at

Here, u is the initial velocity (which is zero), a is the acceleration (90 cm), and t is the time (1 second).

Substituting the values, we have:

v = 0 + (90 cm) * (1 second)
v = 90 cm/second

Therefore, the velocity of the cart after half the time is 90 cm/second.