A car travels along a straight road for 30 seconds starting at time t = 0. Its acceleration in ft/sec2 is given by the linear graph below for the time interval [0, 30]. At t = 0, the velocity of the car is 0 and its position is 10. What is the total distance the car travels in this 30 second interval?
the graph has a y intercept of 10 and an x intercept of 10.
WHy would you add 20 ft
the acceleration graph is apparently
a(t) = 10-t
so, that gives us
v(t) = 10t - t^2/2 + C
v(0)=0, so C=0
so, now we have
s(t) = 5t^2 - t^3/6 + C
s(0) = 10, so C = 10
s(t) = 10 + 5t^2 - t^3/6
s(30) = 0
So, the final position is 10 ft behind the starting line.
However, the total distance traveled is
∫[0,30] |10t - t^2/2| dt = 4000/3 + 20 ft
There's no extra 20 XD
To find the total distance traveled by the car in the 30-second interval, we need to find the area under the graph of the car's velocity.
Since the acceleration of the car is given by a linear graph, we can find the equation of the graph using the given information. We know that at t = 0, the velocity of the car is 0 and its position is 10. The graph has a y-intercept of 10 and an x-intercept of 10.
Using the equation of a line, we can find the equation of the graph. The equation of a line can be written as y = mx + b, where m is the slope of the line.
The slope of the line can be calculated as (change in y)/(change in x), which in this case is (10-0)/(10-0) = 1.
Therefore, the equation of the graph for the car's acceleration is y = x + 10.
To find the total distance traveled, we need to find the area under the graph from t = 0 to t = 30. Since the graph is a straight line, the area can be calculated as the area of a triangle.
The base of the triangle is 30 seconds, and the height is the velocity of the car at t = 30, which we can find using the equation of the graph.
Plugging in t = 30 into the equation y = x + 10, we get y = 30 + 10 = 40. Therefore, the height of the triangle is 40.
The area of a triangle is given by the formula (base * height)/2. Plugging in the values, we get (30 * 40)/2 = 600.
Therefore, the total distance the car travels in the 30-second interval is 600 feet.