A stunt driver wants to make his car jump over 8 cars parked side by side below a horizontal ramp.?

Question

With what minimun speed must he drive off the horixzontal ramp? The vertical height of the ramp is 1.5 m above de cars, adn the horizontal distance he must clear is 20m. If the ramp tilted upward, so that takeoff angle is 10 degrees above the horizontal, what is the new minimum speed?

Answer 1

Let's find how much time it takes to drop by 1.5m.

Taking the vertical angle:

initial vertical speed Viv = 0 m/s (since the ramp is horizontal)
initial height Di = +1.5m
final height D = 0
acceleration a = -9.8 m/s^2

Apply the distance equation

d=1/2at^2 + vt + Di
0 = -4.9 t^2 + 1.5
t = sqrt(1.5/4.9) = 0.55s

Taking the horizontal angle: the car must cross 20m in 0.55s so minimum speed is 20m/0.55s = 36.4 m/s

If the ramp has a 10 degrees angle. The speed now have vertical and horizontal components:

Viv = sin(10)Vi
Vih = cos(10)Vi

Taking the vertical angle:

initial vertical speed Viv = Sin(10)Vi m/s
initial height Di = +1.5m
final height D = 0
acceleration a = -9.8 m/s^2

Apply the distance equation

d=1/2at^2 + vt + Di
0 = -4.9 t^2 + sin(10)Vi.t + 1.5

Taking the horizontal angle: the car must cross 20m in 0.55s so minimum speed is

cos(10)Vi x t = 20 m
Vi = 20/(t*cos(10))

Replacing this in the first equation we get

0 = -4.9 t^2 + sin(10).20/(t*cos(10)).t + 1.5
0 = -4.9 t^2 + 20sin(10)/(cos(10)) + 1.5
0 = -4.9 t^2 + 3.52 + 1.5
0 = -4.9 t^2 + 5.02
t = sqrt(5.02/4.9) = 1.01 s

and since Vi = 20/(t*cos(10))
Vi = 20/(1.01*cos(10)) = 20.06 m/s which is considerably less than 36 m/s.

Answer 2

I don't normally do homework problems for people here but this one is fun :)

Let's assume that the cars being jumped have no HEIGHT, and that there is no air resistance, so that the real question is how fast one has to leave the ramp in order to land 20m away. If the cars have height, then you have to clear that last car!

At that speed "v", in meters per second (m/s) you would both travel 20m and fall 1.5m in a particular time "t". Therefore, the REAL question is how long it takes to fall 1.5m. However long that takes, you need to travel 20m during that time. 20 meters divided by t seconds, then, is your velocity "v".

A falling body accelerates at -9.8m/s/s, therefore its height in meters, after "t" seconds from the start, is given by 1/2(-9.8)t^2.

Set this equal to -1.5 and get:

-1.5 = -4.9t^2 or t^2 = 0.31 (approximately) and thus t = 0.55 seconds.

Therefore, the car must travel 20m in 0.55 seconds, or be going a bit over 36 m/s.

Now, how to deal with the 10 degree takeoff? Simple! Think of it this way: if it wasn't for gravity and falling, the car would go straight out at 10 degrees upward forever. In the previous example, we calculated the time to fall 1.5 meters because thats how high the car would be off the ground at the 20m mark, if it weren't for gravity. For the 10 degree rise, we can use some trigonometry. In this case the tangent function (tangent of an angle is equal to the "opposite" over the "adjacent" side) gives us:

tan(10degrees) = h/20m where "h" is the additional height in meters above 1.5.

0.18 = h/20, so h = 3.5 meters. At 20 meters out, the car would have risen to to 1.5+3.5 = 5.0 meters above the finish line, if it weren't for gravity. Now we go back to the previous method. How long does it take to fall 5.0 meters?

1/2(9.8)t^2=5.0, so t^2=5.0/4.9. t=1.0 second (2 "significant digits")

Therefore the car has to travel 20 meters in 1 second or be going 20 meters per second.

Hope that helps!

Remember - when you report calculations based on physical measurements, your answer can never be more accurate than your original measurement. This is why you have to respect the number of significant digits you have to work with! You have only 2 significant digits to work with in this problem. Answering "36.4" is therefore a WRONG ANSWER. If you want to make certain you clear that last car, and you are limited to 2 significant digits, say "37" !!! :)



Thanks for giving me something to do with my morning coffee :)

Answer 3

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