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 grou

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 motor-cycle engine dies just as he starts up the ramp. He is going 11 m/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 croco-dile food? Justify your answer by calculating the distance he travels through the air after leaving the end of the ramp

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Thiên Hương 2 months 2021-07-31T23:03:14+00:00 1 Answers 2 views 0

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    2021-07-31T23:04:49+00:00

    Answer:

    He becomes a croco-dile food

    Explanation:

    From the question we are told that

        The height is h = 2.0 m

          The angle is  \theta = 20^o

         The distance is  w = 10m

           The speed is  u = 11 m/s

           The coefficient of static friction is  \mu = 0.02

    At equilibrium the forces acting on the motorcycle are mathematically represented as

            ma = mgsin \theta  + F_f

    where  F_f is the frictional force mathematically represented as

                F_f =\mu F_x =\mu mgcos \theta

    where F_x is the horizontal component of the force

    substituting into the equation

                ma = mgsin \theta  + \mu mg cos \theta

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

                   making  a the subject of the formula

          a = g(sin \theta = \mu cos \theta )

              substituting values

          a = 9.8 (sin(20) + (0.02 ) cos (20 ))

            = 3.54 m/s^2

    Applying ” SOHCAHTOA” rule we mathematically evaluate that length of the ramp as  

                 sin \theta = \frac{h}{l}

    making l the subject

              l = \frac{h}{sin \theta }

    substituting values

            l = \frac{2}{sin (20)}

               l = 5.85m

    Apply Newton equation of motion we can mathematically evaluate the  final velocity at the end of the ramp  as

          v^2 =u^2 + 2 (-a)l

      The negative a means it is moving against gravity

          substituting values

          v^2 = (11)^2 - 2(3.54) (5.85)

               v= \sqrt{79.582}

                  = 8.92m/s

    The initial velocity at the beginning of the pool (end of ramp) is composed of two component which is

        Initial velocity along the x-axis which is mathematically evaluated as

              v_x = vcos 20^o

           substituting values

             v_x = 8.92 * cos (20)

                  = 8.38 m/s

    Initial velocity along the y-axis which is mathematically evaluated as

                 v_y = vsin\theta

          substituting values

                 v_y = 8.90 sin (20)

                      = 3.05 m/s

    Now the motion through the pool in the vertical direction can mathematically modeled as

            y = y_o +  u_yt + \frac{1}{2} a_y t^2

    where y_o is the initial height,

             u_y is the initial velocity in the y-axis

        a_y  is the  initial  acceleration in the y axis  with a constant value of (g = 9.8 m/s^2)

    at the y= 0 which is when the height above ground is zero

          Substituting values

                  0 = 2 + (3.05)t - 0.5 (9.8)t^2

    The negative sign is because the acceleration is moving against the motion

                     -(4.9)t^2 + (2.79)t + 2m = 0

       Solving using quadratic formula

                  \frac{-b \pm \sqrt{b^2 -4ac} }{2a}

    substituting values

                 \frac{-3.05 \pm \sqrt{(3.05)^2 - 4(-4.9) * 2} }{2 *( -4.9)}

                    t = \frac{-3.05 + 6.9}{-9.8}   \ or t = \frac{-3.05 - 6.9}{-9.8}

                    t = -0.39s  \ or  \ t = 1.02s

    since in this case time cannot be negative

                 t = 1.02s

    At this time the position the  motorcycle along the x-axis is mathematically evaluated as

                  x = u_x t

                   x  =8.38 *1.02

                       x =8.54m

    So from this value we can see that the motorcycle would not cross the pool as the position is less that the length of  the pool

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