Question:

A car is moving along a straight line with an initial velocity of 15 m/s. The car accelerates uniformly for a distance of 200 meters and reaches a final velocity of 35 m/s. Determine the acceleration of the car.

(a) What is the acceleration of the car? (b) How much time does it take for the car to reach the final velocity? (c) What is the average velocity of the car during this acceleration period?

Assume all values are positive unless otherwise stated and use standard units for measurement.

Answer:

(a) To find the acceleration of the car, we can use the formula:

final velocity (Vf) = initial velocity (Vi) + (acceleration (a) * time (t))

Given:

Vi = 15 m/s
Vf = 35 m/s
Distance (d) = 200 m

We can rearrange the formula to solve for acceleration (a):

Vf = Vi + (a * t)
35 = 15 + (a * t)

We also know that the car travels a distance of 200 meters during this acceleration period. We can use the formula:

d = Vi * t + (1/2) * a * t^2

Substituting the given values:

200 = 15t + (1/2) * a * t^2

Now we have two equations with two variables (a and t). We can solve for acceleration using these equations.

First, rearrange the equation for distance to solve for time (t):

200 = 15t + (1/2) * a * t^2
200 = 15t + (1/2) * at^2

Since the equation is quadratic, we can rearrange it to:
at^2 + 30t - 400 = 0

Using the quadratic formula:

t = (-b ± sqrt(b^2 - 4ac))/(2a)

a = 1, b = 30, c = -400

t = (-30 ± sqrt(30^2 - 4 * 1 * -400))/(2 * 1)
t = (-30 ± sqrt(900 + 1600))/2
t = (-30 ± sqrt(2500))/2
t = (-30 ± 50)/2

Since time cannot be negative, we take the positive value:
t = (-30 + 50)/2
t = 20/2
t = 10 s

Now that we have the value of time (t), we can substitute it back into the first equation to find the acceleration (a):

35 = 15 + (a * 10)
20 = 10a
a = 2 m/s^2

Therefore, the acceleration of the car is 2 m/s^2.

(b) The time it takes for the car to reach the final velocity is 10 seconds.

(c) The average velocity of the car during this acceleration period can be calculated using the formula:

Average velocity = (Vi + Vf) / 2

Substituting the given values:

Average velocity = (15 + 35) / 2
Average velocity = 50 / 2
Average velocity = 25 m/s

Therefore, the average velocity of the car during this acceleration period is 25 m/s.