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 velocity of the car when t = 6?
I can't show the graph, but the y intercept is at (0,10) and the x intercept is (10,0).
Can someone help me get started on this problem? Then I think I can do the rest.
To find the velocity of the car at t = 6, we need to determine its acceleration at that time and integrate it over the given time interval.
From the given linear graph, we can determine the equation of the line that represents the acceleration. Since two points on the line are (0,10) and (10,0), we can use the slope-intercept form of the equation for a line:
y = mx + b
where m is the slope and b is the y-intercept.
First, let's find the slope (m):
m = (change in y) / (change in x) = (0 - 10) / (10 - 0) = -10/10 = -1
Now, we have the equation for acceleration:
a = -t + b
where t is the time in seconds and b is the y-intercept (in this case, b = 10).
Next, we need to integrate the acceleration equation to find the velocity equation:
v = ∫a dt
Integrating the equation with respect to t:
v = -∫t dt + ∫b dt
v = -0.5t^2 + bt + C
where C is the constant of integration.
To find the constant of integration, we can use the initial conditions given in the problem. At t = 0, velocity (v) is 0, and the position (s) is 10. Since velocity is the derivative of position with respect to time, we can find the constant of integration by substituting these values into the velocity equation:
0 = -0.5(0)^2 + b(0) + C
0 = 0 + 0 + C
C = 0
Now we have the equation for velocity:
v = -0.5t^2 + 10t
To find the velocity at t = 6, substitute t = 6 into the equation:
v = -0.5(6)^2 + 10(6)
v = -0.5(36) + 60
v = -18 + 60
v = 42 ft/sec
Therefore, the velocity of the car at t = 6 is 42 ft/sec.
Apparently a(t) = 10-t
the velocity v(t) = ∫a(t) dt
use v(0)=0 to determine the C introduced
the position s(t) = ∫v(t) dt
use s(0)=10 to determine the C introduced.