Question

(a) A daredevil is attempting to jump his motorcycle over a line of buses parked end to end by driving up a 32º ramp at a speed of 40.0 m/s (144 km/h) . How many buses can he clear if the top of the takeoff ramp is at the same height as the bus tops and the buses are 20.0 m long? (b) Discuss what your answer implies about the margin of error in this act—that is, consider how much greater the range is than the horizontal distance he must travel to miss the end of the last bus. (Neglect air resistance.)

Answer 1

a) 7 buses

b) 6.8 m

Explanation:

a)

The motion of the motorcycle is a projectile motion, so it consists of 2 independent motions:

- A uniform motion along the horizontal direction

- A uniformly accelerated motion along the vertical direction

The initial components of the velocity of the motorcycle are:

$v_x = u cos(32^{\circ})=(40.0)(cos 32^{\circ})=33.9 m/s\\v_y = u sin(32^{\circ})=(40.0)(sin 32^{\circ})=21.2 m/s$

The equation for the vertical motion of the motorcycle is

$y=h+u_y t - \frac{1}{2}gt^2$

where

y is the altitude at time t

h is the initial height

$g=9.8 m/s^2$ is the acceleration due to gravity

The bus top is at the same height of the initial ramp, so we have

$y=h$

And therefore, we can solve the equation for t, to find the time of flight:

$0=u_y t - \frac{1}{2}gt^2\\t(u_y-\frac{1}{2}gt)=0\\t=\frac{2u_y}{g}=\frac{2(21.2)}{9.8}=4.33 s$

Now we find what is the horizontal distance covered by the motorcycle in its jump, which is given by:

$d=v_x t = (33.9)(4.33)=146.8 m$

And since each bus has a length of L = 20.0 m, the number of buses that the motorcycle can clear with its jump is:

$n=\frac{d}{L}=\frac{146.8}{20}=7.34$

So, 7 buses.

b)

In the previous problem, we saw that the total range of the motion of the motorcycle is

$d=146.8 m$

And we said that this corresponds to 7 buses.

Each bus has a length of

L = 20 m

So, the total length of 7 buses is

$L' = 7L=7(20)=140 m$

Therefore, the range of the motorcycle is greater than the length of the buses by:

$\Delta x = d-L'=146.8-140 = 6.8 m$

which means he will miss the last bus by 6.8 meters.