Question

A motorcycle daredevil plans to ride up a $2.0-m$-high, $20°$ramp, sail across a $10-m$-wide pool filled with hungry crocodiles, and land at ground level on the other side. He has done this stunt many times and approaches it with confidence. Unfortunately, the motorcycle engine dies just as he starts up the ramp. He is going $11m/s$at that instant, and the rolling friction of his rubber tires (coefficient $0.02$) is not negligible. Does he survive, or does he become crocodile food? Justify your answer by calculating the distance he travels through the air after leaving the end of the ramp.

Short answer

The range$8.56\mathrm{m}$ is less than the width of the pool, the biker will land on the pool and hence, will not survive.

Step-by-step solution

Given Information

We know that the angle of inclination is $\theta =20°$, height of the inclination is $h=2.0\mathrm{m}$, the coefficient of friction between the bike tire and the inclination surface is $\mu =0.02$, width of the pool is $d=10.0\mathrm{m}$, acceleration due to gravity $g=9.81\mathrm{m}/{\mathrm{s}}^2$ and the initial speed of the bike is $v_{\mathrm{mtight}}>\mathrm{i}=11.0\mathrm{m}/\mathrm{s}$. We have to calculate the distance of landing position from the end of inclination.

Apply the equation of force motion

Let's consider the information given in the question:

The equation force motion is given as

$ma=mg\sin \theta +F_f$

Where,

$F_f$is the friction force

$ma=mg\sin \theta +\mu mg\cos \theta$

$ma=mg(\sin \theta +\mu \cos \theta$

$\Rightarrow a=g(\sin \theta +\mu \cos \theta )$

Substitute the values 

Substitute the values

$a=(9.8m/s^2)(\sin \theta {20}^0+(0.02\left)\cos {20}^0\right)$

$a=(9.8m/s^2)(0.36)$

$a=3.54m/s^2$

Therefore, the length of the ramp is given as,

$\sin \theta =\frac{h}{L}$

Where,

$h$is the height$=2m$

$L$is the length of the ramp

So,

$L=\frac{h}{Sin\theta }$

$\Rightarrow L=\frac{2m}{Sin{20}^0}$

$L=5.85m$

Apply the equation of motion

Let's consider the equation of motion:

$v^2=u^2+2(-a)L$

where,

$v$is the initial velocity

$u$is the instant velocity

$v^2=u^2-2al$

role="math" localid="1648361146542" $v^2={(11m/s)}^2-2(3.54m/s^2)(5.85m)$

role="math" localid="1648361542478" $v^2=79.58m^2/s^2$

role="math" localid="1648361565005" $\Rightarrow v=\sqrt{79.58m^2/s^2}$

role="math" localid="1648361593999" $v=8.92m/s$

Calculate the initial velocity of two component

The length of the pool is $10m$.

Calculate the initial velocity has two components along the horizontal direction and the vertical direction as given as,

$v_x=v\cos {20}^0$

$v_x=(8.92m/s)\cos {20}^0$

$v_x=8.38m/s$

Compute for $v_y$

$v_y=v\sin {20}^0$

$v_y=(8.92m/s)\sin {20}^0$

$v_y=3.05$

Therefore, the equation along the vertical direction is given as:

$y=y_0+u_{y}t+\frac{1}{2}{a}_y{t}^2$

$0=2\mathrm{m}+(3.05\mathrm{m}/\mathrm{s})t-\frac{1}{2}\left(9.8\mathrm{m}/{\mathrm{s}}^2\right)t^2$

role="math" localid="1648363560106" $\Rightarrow -\left(4.9\mathrm{m}/{\mathrm{s}}^2\right)t^2+(3.05\mathrm{m}/\mathrm{s})t+2\mathrm{m}=0$

On solving we get,

$t=1.02$

Calculate the position of the motor cycle

The position of the motorcycle during the projectile time is given as,

$x=u_{x}t$

$x=(8.38\mathrm{m}/\mathrm{s})(1.02\mathrm{s})$

$x=8.547$

Here we see that the distance or position of the motorcycle is less than the length of the pool. So, the motorcycle cannot cross the pool and he falls into the pool.

Final Answer 

The distance or position of the motorcycle is 8.54m and it is less than the length of the pool. So, the motorcycle cannot cross the pool and he falls into the pool