A daredevil tries to jump a canyon of width 15 m. To do so, his motorcycle up an incline slope at an angle of 22 degrees. What minimum initial velocity is necessary to clear the canyon?
it will take 15/(v cos22°) seconds to cross the gap.
It will take half that time to reach the top of the parabola, so find when
the vertical speed is v sin22° - 9.8t = 0
Superman can “leap tall buildings in a single bound.” Assuming the man of steel starts and stops on the ground, how high a building could superman jump over if you want to leave the ground at the speed of 82.0 m/sec at a 75° launch angle.
To find the minimum initial velocity necessary for the daredevil to clear the canyon, we can use the principle of conservation of energy. The initial potential energy of the daredevil-motorcycle system at the bottom of the incline is converted into kinetic energy as the daredevil approaches the edge of the canyon. To clear the canyon, the kinetic energy at the edge of the canyon should be equal to the initial potential energy.
Let's break down the problem step by step:
Step 1: Calculate the height of the incline
The height of the incline can be found by using trigonometry. As we know the width of the canyon and the angle of the incline, we can use the sine function:
sin(θ) = opposite / hypotenuse
sin(22°) = height / width
height = width * sin(22°)
Step 2: Calculate the potential energy at the bottom of the incline
The potential energy (PE) of an object near the Earth's surface can be calculated using the formula:
PE = mass * gravitational acceleration * height
Since the daredevil and the motorcycle are treated as a system, their mass cancels out, and we can use the simplified formula:
PE = gravitational acceleration * height
Step 3: Calculate the kinetic energy at the edge of the canyon
The kinetic energy (KE) of an object can be calculated using the formula:
KE = (1/2) * mass * velocity^2
Since the mass of the daredevil and the motorcycle is treated as a system, we can use the simplified formula:
KE = (1/2) * velocity^2
Step 4: Set up the conservation of energy equation
As mentioned earlier, for the daredevil to clear the canyon, the kinetic energy at the edge of the canyon should be equal to the initial potential energy:
(1/2) * velocity^2 = gravitational acceleration * height
Step 5: Solve for the minimum initial velocity (velocity^2)
To get the minimum initial velocity, we can rearrange the equation:
velocity^2 = 2 * gravitational acceleration * height
Step 6: Substitute the known values and solve
Substitute the known values into the equation:
velocity^2 = 2 * 9.8 m/s^2 * (15 m * sin(22°))
velocity^2 = 2 * 9.8 m^2/s^2 * (15 m * 0.3746)
velocity^2 = 2 * 9.8 m^2/s^2 * 5.619 m
velocity^2 = 109.2246 m^2/s^2
Finally, take the square root to find the minimum initial velocity:
velocity ≈ √(109.2246 m^2/s^2)
velocity ≈ 10.453 m/s
Therefore, the daredevil needs a minimum initial velocity of approximately 10.453 m/s to clear the canyon.