A student accelerates from rest at a rate of 3 miles/min^2. How far will the car have traveled at the moment it reaches a velocity of 65 miles/60 min?
a(t) = 3 miles/min^2
so v(t) = 3t + k miles/min
but when t=0, v=0 so k=0
then v(t)=3t
d(t) = 3/2 t^2 + c
but when t=0 d, the distance, equals zero
so c=0
so you have a(t)=3, v(t)=3t, d(t)=3/2 t^2
watch your units
when vel=65 miles/60minutes is the same as 13/12 miles/min
then 3t=13/12
t=13/36 minutes
at that t, d = 3/2 (13/12)^2 = 1.76 miles
by the way, since 1 mile/min = 60 mph, after 2 seconds, the velocity would have been 6 miles/min or 180 mph, after 3 seconds 9 miles/min or 540 mph.
Where do I invest in this "car", I want to enter it in a Formula I race.
To determine how far the car will have traveled at the moment it reaches a velocity of 65 miles/60 min, we can use the equations of motion.
First, let's analyze the equations provided:
- The acceleration equation is given as a(t) = 3 miles/min^2.
- The velocity equation is derived from integrating the acceleration equation: v(t) = 3t + k miles/min. Since the car starts from rest (t = 0, v = 0), the integration constant k is 0.
- The distance equation is derived from integrating the velocity equation: d(t) = (3/2)t^2 + c. Since the initial distance is also 0, the integration constant c is 0 as well.
We now have the equations: a(t) = 3, v(t) = 3t, and d(t) = (3/2)t^2.
To find the time at which the velocity is 65 miles/60 min, we equate the velocity equation with the given value:
3t = 65/60
Solving for t, we find:
t = (65/60) / 3
t = 13/36 minutes
Now, plug in this value of t into the distance equation:
d(t) = (3/2)(13/36)^2
d(t) = 1.76 miles
Therefore, at the moment the car reaches a velocity of 65 miles/60 min, it will have traveled a distance of 1.76 miles.
However, it's important to note that the units for velocity are not consistent with those typically used for car speeds. 65 miles in 60 minutes is equivalent to 65 miles per hour, which is a common unit for measuring car speeds. The given acceleration and velocity units do not align with traditional automotive units, so caution should be exercised when interpreting the results.
Also, keep in mind that the equations used here are simplified kinematics equations and may not accurately represent real-world scenarios. Investing in a Formula I race car should involve considering other factors such as handling, aerodynamics, and driver skill.