The maker of an automobile advertises that it takes 10 seconds to accelerate from 30 kilometers per hour to 90 kilometers per hour. Assuming constant acceleration, compute the following
distance traveled in 10 seconds
Position,(distance) is the 2nd antiderivative of acceleration, so first you need to make an equation for acceleration, being 90=30+6t.
From there you would have to antiderivate the equation twice, using an intial value for velocity and position to solve for the C's that are created when taking the antiderivative of something, so without that information I can't help you. When you get that you would need to simply plug in 10 for your t value and there is your answer
a = (90-30)/10 = 6 km/hr-s
s = 1/2 at^2 = 3 km/hr-s * (10s)^2
= 300 km-s/hr
= 300 km-s/hr * 1hr/3600s
= 300/3600 km
= 1/12 km
or, since you want to use calculus,
a = 6km/hr-s
= 6km/hr-s * 1hr/3600s
= 1/600 km/s^2
so, a(t) = 1/600
v(t) = 1/600 t
s(t) = 1/1200 t^2
s(10) = 100/1200 = 1/12 km
To compute the distance traveled in 10 seconds, we need to use the equation of motion for uniformly accelerated motion:
\[
d = ut + \frac{1}{2}at^2
\]
Where:
- \(d\) is the distance traveled
- \(u\) is the initial velocity
- \(a\) is the acceleration
- \(t\) is the time taken
In this case, the initial velocity (\(u\)) is 30 kilometers per hour (since the car starts from 30 km/h), the final velocity is 90 kilometers per hour, and the time taken (\(t\)) is 10 seconds.
However, we need to convert the velocities from kilometers per hour to meters per second because the formula requires it in SI units.
To convert kilometers per hour to meters per second, we need to multiply by a conversion factor of 1000/3600 (since there are 1000 meters in a kilometer and 3600 seconds in an hour).
Thus, we have:
Initial velocity (\(u\)) = 30 km/h \(\times\) (1000 m/km) / (3600 s/h) = 8.33 m/s
Final velocity (\(v\)) = 90 km/h \(\times\) (1000 m/km) / (3600 s/h) = 25 m/s
Plugging in these values into the equation of motion, we get:
\[
d = (8.33 \, \text{m/s})(10 \, \text{s}) + \frac{1}{2} a (10 \, \text{s})^2
\]
To find the distance traveled, we need to know the acceleration (\(a\)) of the car. Unfortunately, the question does not provide this information. Without the acceleration, we cannot determine the distance traveled accurately.