Motion in 2 and 3 dimentions

[Beretta]

Carlos is on his trail bike, approaching a creel bed that is 7m wide. A ramp with an incline of 10° has been built for daring people who try to jump the creek. Carlos is traveling at his bike’s maximum speed, 40km/h. a) Should Carlos attempt the jump or emphatically hit the brakes? b) What is the minimum speed a bike must have to make the jump? Assume equal elevations on either side of the creek.


Velocity = 40km/h = 11.11m/s

vx = v cos theta = 11.11m/s cos 10° =10.94m/s
vy = v sin theta = 11.11 m/s sin 10° = 1.93m/s
(Not sure for example here if I should consider significant figure on each calculation or only in the final answer)

a)
To obtain time Carlos spends in the air:
Y(t) = y0 + vy(t) -1/2(g)(t^2)
(Is it only when Y is positive upward and objects are falling downwards g should be negative?)
0m = 0m + 1.93t – (4.905m/s^2)
t(1,93m/s – (4,905m/s^2)t )

t = 0s and t =(1.93m/s)/(4.905m/s^2) = 0.40s (Again not sure if I should round or not)

To obtain distance:
X(t) = x0 + v0x(t)
X(t) = 10.94m/s(0.40s) = 4.4m and Carlos should hit the brakes!

b)
To calculate the minimum speed:
Vertical Time = 2v0y/g = 2v0 sin theta / g
And then I put time in the horizontal equation thus:
X(t) = x0 + v0x(t)
7m = 0m + v0x (2v0 sin theta / g)
7m = (v0 cos theta) (2v0 sin theta / g)
7m = (v0 cos 10) ((2v0 sin 10)/9.81)
7m = 2(v^2)(0,1666)/9,81
v^2 = 68.67/0.332 = 206.8(m/s)^2
v = 14.4m/s

Is my reasoning right? I’m really in need of your comments, and thank you in advance.

[HallsofIvy]

Looks good to me. As for your specific questions:
"(Not sure for example here if I should consider significant figure on each calculation or only in the final answer)"
As a general rule, include one more figure in your calculations than you put in your answer. In this case, you are given, at best, 2 significant figures so you need 2 in your answer.

"(Is it only when Y is positive upward and objects are falling downwards g should be negative?)"
A coordinate system is something you impose on the problem yourself. What most people do- and I recommend- is take y to be positive upward (so "falling down" is negative) and g negative.

You calculated the distance Carlos will go before "touching down" again and found that it was less than the distance across the creek.
Okay, he shouldn't try it. I see no problem with your calculation.

You finally calculated the speed necessary so that horizontal dstance is at least 7 m. Once again, it looks good to me.

Finally, you calculated the minimum speed necessary to clear the creek and it was slightly MORE than Carlo's actual speed.

[Beretta]

Thank you very much, I appreciate your comment.