Question

A copy-cat daredevil tries to reenact Evel Knievel's 1974 attempt to jump the Snake River Canyon in a rocket-powered motorcycle. The canyon is \(L=400 . \mathrm{m}\) wide, with the opposite rims at the same height. The height of the launch ramp at one rim of the canyon is \(h=8.00 \mathrm{~m}\) above the \(\mathrm{rim},\) and the angle of the end of the ramp is \(45.0^{\circ}\) with the horizontal.

Short answer

Answer: To determine if the daredevil makes it to the other side of the canyon for any given initial velocity \(v_0\), we should look at the height of the daredevil \(y\) when reaching the other side. If \(y\geq 0\), the daredevil successfully crosses the canyon. The height at the other side is given by: \(y=h+\left(v_0\sin{45^\circ}\right)\left(\frac{L}{v_0\cos{45^\circ}}\right)-\frac{1}{2}g\left(\frac{L}{v_0\cos{45^\circ}}\right)^2\). This analysis will help in determining the outcome of the jump based on given initial velocity \(v_0\).

Step-by-step solution

Determine the initial velocity of the daredevil

We are given that the ramp has a height of \(h=8.00\,\text{m}\) and an angle of \(45.0^\circ\) with the horizontal. To find the initial velocity of the daredevil, we can use trigonometry to find the velocity components in the horizontal and vertical directions. Let \(v_0\) be the initial velocity of the daredevil. The horizontal component of the initial velocity can be found using the following equation: \(v_{0x}=v_0\cos{45^\circ}\) The vertical component of the initial velocity can be found using the following equation: \(v_{0y}=v_0\sin{45^\circ}\) Note: The problem does not provide information about the initial velocity (\(v_0\)), so we will leave it as a variable in the solution.

Determine the time needed to cross the canyon

The time needed to cross the canyon can be determined by analyzing the horizontal motion of the daredevil. Since there is no acceleration in the horizontal direction, we can use the following equation to find the time of the flight (\(t\)): \(L = v_{0x} \cdot t\). Substituting \(v_{0x}=v_0\cos{45^\circ}\) and rearranging for \(t\), we get: \(t=\frac{L}{v_0\cos{45^\circ}}\).

Determine the height of the daredevil when reaching the other side of the canyon

To find the height (\(y\)) at which the daredevil reaches the other side of the canyon, we can analyze the vertical trajectory using the following equation for projectile motion: \(y=h + v_{0y}\cdot t -\frac{1}{2}gt^2\). Substituting \(v_{0y} =v_0\sin{45^\circ}\) and \(t=\frac{L}{v_0\cos{45^\circ}}\) in the equation, we get: \(y=h+\left(v_0\sin{45^\circ}\right)\left(\frac{L}{v_0\cos{45^\circ}}\right)-\frac{1}{2}g\left(\frac{L}{v_0\cos{45^\circ}}\right)^2\).

Determine if the daredevil reaches the opposite rim

To find out if the daredevil successfully crosses the canyon, we need to check if the height \(y\) at the other side of the canyon is greater than or equal to zero. If \(y\geq 0\), then the daredevil has made it to the other side. This analysis is based on the given parameters and can be used to determine if the daredevil makes the jump for any given initial velocity \(v_0\).

Key concepts

Kinematics

Kinematics is the branch of physics concerned with the motion of objects without considering the forces that cause the motion. When studying the motion of projectiles, like our ramp-jumping daredevil, we focus on two key components: the velocity of the projectile and its position over time.

In our scenario, the daredevil's rocket-powered motorcycle moves through the air under the influence of gravity, following a parabolic path known as the trajectory. By using the principles of kinematics, we can break down this trajectory into two simpler motions: horizontal and vertical. The horizontal motion occurs at constant velocity since there is no acceleration (assuming air resistance is negligible), while the vertical motion is subject to constant acceleration due to gravity.

Essential kinematic equations help us analyze the motion of the projectile. For example, the horizontal distance covered by the daredevil can be predicted using the equation for constant velocity motion, while the vertical height can be determined using the equations accounting for constant acceleration. These kinematic principles fundamentally underpin the step-by-step solution to our textbook problem.

Trajectory Analysis

Trajectory analysis involves examining the path that a projectile follows through space, which, under the influence of gravity alone, is typically a parabola. By decomposing the daredevil's jump into its horizontal and vertical components, we can predict the entire path of the projectile.

The launch angle, in this case, 45 degrees, plays a crucial role in determining the shape and reach of the trajectory. At an angle of 45 degrees, the daredevil's jump will have equal horizontal and vertical initial velocity components, optimizing the range of the jump under ideal conditions.

We can calculate the necessary initial velocity (\( v_0 \)) that ensures the daredevil will land safely on the other side by plugging in the values into the motion equations for the vertical and horizontal components. However, to ensure that the projectile lands at the correct height, the vertical velocity, gravitational force, and time in the air must be carefully considered alongside the horizontal motion.

Horizontal and Vertical Velocity Components

The motion of the daredevil's jump is greatly influenced by its initial velocity components. The initial horizontal velocity (\(v_{0x}\)) affects how far the daredevil will travel, while the initial vertical velocity (\(v_{0y}\)) will influence the peak height reached during the jump.

These velocity components are derived from the initial launch speed and the angle of the ramp. Using trigonometric functions, we can determine that the horizontal component is \(v_{0}\times\text{cos}(45^\text{o})\) and the vertical component is \(v_{0}\times\text{sin}(45^\text{o})\text{,}\) given the 45-degree angle.
For our scenario, it's the interplay of these components that dictates whether the jump succeeds or fails. The horizontal component dictates how long the daredevil will be in the air, determined by the canyon's width, while the vertical component—working against gravity—determines if he’ll reach the height of the opposite rim. The balance of these components is vital for completing the jump successfully.