A car is stopped at a red light, and starts moving at time t = 0 when the light turns green, with an acceleration given by a(t) = √t, with t given in seconds, and distances given in metres.

a) Find a formula for the velocity v(t), via integration.

b) How long will it take for this car to travel 100 metres, from the moment it started moving?

a) To find the formula for velocity v(t), we need to integrate the given acceleration function a(t) = √t with respect to time.

∫ a(t) dt = ∫ √t dt

Using the power rule for integration, we can rewrite the integral as:

∫ √t dt = ∫ t^(1/2) dt

Applying the power rule, we add 1 to the exponent and divide by the new exponent:

= (2/3) * t^(3/2) + C

where C is the constant of integration.

Therefore, the formula for velocity v(t) is:

v(t) = (2/3) * t^(3/2) + C

b) To determine how long it will take for the car to travel 100 meters, we need to set up an equation using the distance formula.

The distance travelled d(t) at any time t can be found by integrating the velocity function v(t):

d(t) = ∫ v(t) dt

Using the velocity function derived above:

d(t) = ∫ [(2/3) * t^(3/2) + C] dt

Integrating each term separately:

d(t) = [(2/3) * (2/5) * t^(5/2)] + C * t + K

where K is another constant of integration.

Since the car starts at time t = 0, we can determine the value of C. At t = 0, the distance d(0) is 0, so:

d(0) = [(2/3) * (2/5) * (0)^(5/2)] + C * 0 + K

0 = 0 + 0 + K

Therefore, K = 0.

Now we have the equation for distance:

d(t) = (4/15) * t^(5/2)

To find the time it takes for the car to travel 100 meters, we set up the equation:

100 = (4/15) * t^(5/2)

Solving for t requires some manipulation of the equation:

t^(5/2) = (15/4) * 100

t^(5/2) = 375

Taking the square root of both sides:

t = (375)^(2/5)

Using a calculator or simplifying further, we get:

t ≈ 5.524 seconds

Therefore, it will take approximately 5.524 seconds for the car to travel 100 meters from the moment it started moving.