Describe the velocity of the motorbike from 2seconds to 8seconds
Answers
Any distance in the set x∈(0,c∗10) is a possible answer. To be more specific, you have to add at least one more condition, such as with constant acceleration. You might also want to specify that the motorcycle is using its own engine and is not in a different inertial reference frame.
These aren’t trivial additions. Motorcycles generally do not produce constant acceleration — they are more likely to produce constant power when you crank them up, and then there is the shifting of gears. In intro physics classes, of course, one usually just ignores all of this to get the student to focus on “simple” constant acceleration kinematics. Guessing that I’m doing your physics homework for you (tut! tut! for shame! do it yourself!) I’ll give the very simplest answer for constant acceleration:
a=20−810=1.2 m/sec 2 .
Δx=v2f−v202a
(kinematic relation found by solving equations of motion and eliminating time.) Plugging in:
Δx=400−642.4=336/2.4=140 meters
(assuming I didn’t make any trivial arithmetic errors, always possible).
For grins, suppose we have constant power instead and no gear shifts or friction or drag and that we know the mass m of the motorcycle (which obviously matters). Then algebraically:
P0=mv2f−mv202Δt=Fv
throughout the motion. (This is the total work done divided by the total time for constant power, which is also the force times the speed in the direction of the force.) P0 is a computable number (if we know the mass), so we can solve for the force:
F=P0v=ma=mdvdt
This is a very different result than we get for constant acceleration, and yet is entirely plausible — a lot more plausible than a motorcycle that delivers a constant forward directed force, actually! Now we have to solve for v(t) (and it turns out that the pesky m doesn’t matter after all).
vdv=P0mdt
∫vv0vdv=12(v2−v20)=P0m∫t0dt=P0tm
This yields:
dxdt=v(t)=v20+2P0tm−−−−−−−√
which can be integrated, with a tiny bit of pain, to find Δx.
I’d do the integral, but I’m out of time, so consider it an exercise to make up for the fact that I did your homework for you.