Question

Your friend's car is parked on a cliff overlooking the ocean on an incline that makes an angle of \(17.0^{\circ}\) below the horizontal. The brakes fail, and the car rolls from rest down the incline for a distance of \(29.0 \mathrm{~m}\) to the edge of the cliff, which is \(55.0 \mathrm{~m}\) above the ocean, and, unfortunately, continues over the edge and lands in the ocean. a) Find the car's position relative to the base of the cliff when the car lands in the ocean. b) Find the length of time the car is in the air.

Short answer

a) Use the method described in step 6 to find the car's final position relative to the base of the cliff. b) Use the method described in step 4 to find the time the car spends in the air.

Step-by-step solution

Identify the known information

: The given information in the problem is: 1. Incline angle: \(17.0^{\circ}\) 2. Distance along the incline: \(29.0~\mathrm{m}\) 3. Height of the cliff: \(55.0~\mathrm{m}\)

Calculate the initial velocity

: To find the initial velocity (\(v\)) at which the car goes off the cliff, we need to consider its motion along the incline. We can use the kinematic equations for uniformly accelerated linear motion. The initial velocity \(v_0 = 0\) (since it starts from rest). Acceleration along the incline is \(a = g\sin(17.0^{\circ})\), where \(g\) is the acceleration due to gravity (\(9.81~\mathrm{m/s^2}\)). Using the equation \(v^2 = v_0^2 + 2ad\), we get: \(v^2 = 0^2 + 2(9.81\sin(17.0^{\circ}))(29.0)\)

Calculate the initial horizontal velocity (\(v_{ix}\)) and initial vertical velocity (\(v_{iy}\))

: We need to find the components of the initial velocity in the horizontal and vertical directions. For this, we use trigonometry. \(v_{ix} = v\cos(17.0^{\circ})\) \(v_{iy} = v\sin(17.0^{\circ})\)

Calculate the time of flight of the car

: Now we move on to the parabolic trajectory of the car as it goes off the cliff. The time the car is in the air will be determined by the vertical motion. The cliff has a height of \(55.0~\mathrm{m}\). We can use the equation \(y_{final} = y_{initial} + v_{iy}t - \frac{1}{2}gt^2\) (where \(y_{final}\) is the final height, \(y_{initial}\) is the initial height, \(v_{iy}\) is the initial vertical velocity, and \(t\) is the time). Since the car lands in the ocean, \(y_{final} = 0\). Setting \(y_{initial}\) to be the height of the cliff, the equation becomes: \(0 = 55.0 + v_{iy}t - \frac{1}{2}(9.81)t^2\) Solve the equation for \(t\). This will give two possible values of \(t\), one positive and one negative. The negative value is not physically meaningful, so keep the positive value of \(t\).

Calculate the horizontal distance traveled by the car

: Finally, we need to find the horizontal distance the car travels in the air. The horizontal motion is constant, with a velocity of \(v_{ix}\). Using the equation \(x = v_{ix}t\), we get the horizontal distance traveled by the car, \(x\).

Calculate the final position relative to the base of the cliff

: Now we have everything we need to find the position of the car relative to the base of the cliff when it lands. To do this, we will have to find the horizontal distance traveled by the car while moving on the incline. For this, we need to find the horizontal component of the \(29.0~-m\) distance along the incline, which is given by \(29.0\cos(17.0^{\circ})\). Add this distance to the horizontal distance traveled during free fall, and this will give the final position of the car relative to the base of the cliff. a) The car's position relative to the base of the cliff when the car lands in the ocean is found in step 6. b) The length of time the car is in the air is found in step 4.

Key concepts

Kinematic Equations

Kinematic equations are the foundation of classical mechanics and provide essential tools for solving problems related to the motion of objects. These equations describe the relationships between displacement, velocity, acceleration, and time without considering the underlying causes of the motion, such as forces. In our projectile motion problem, we apply these equations to determine the initial speed at which the car leaves the incline and its subsequent flight time. By using the equation

\(v^2 = v_0^2 + 2ad\)

with initial velocity \(v_0 = 0\) and acceleration \(a = g\sin(\theta)\), we can find the speed at which the car rolls off the edge. By splitting this speed into horizontal and vertical components, we are ready to analyze the projectile motion of the car once it is airborne. Understanding kinematic equations equips students with a powerful method for dissecting complex motion into manageable calculations.

Inclined Plane Motion

The motion on an inclined plane deals with an object's movement on a surface that is tilted with respect to the horizontal. This type of motion is significant because it introduces components of gravitational force acting along and perpendicular to the surface of the incline. For our rolling car, the gravitational pull down the incline is characterized by the force component
\(g\sin(\theta)\)
which causes the car to accelerate. By accounting for the incline angle of \(17.0^\circ\), the gravitational force is no longer acting solely in the vertical direction, but partly pushing the car along the plane.

In the kinematic analysis, it's important to consider this component when using equations to predict the object's velocity after traversing a known distance on the incline. This step is essential before the car reaches the cliff's edge and becomes a projectile.

Parabolic Trajectory

When an object is projected into the air and is only subjected to the force of gravity, it follows a parabolic trajectory. This concept is beautifully demonstrated by a car in our problem, flying off the edge of a cliff into the ocean. Its horizontal motion is characterized by a constant velocity, given there's no air resistance, while its vertical motion is influenced by the acceleration due to gravity. By calculating the time in the air

\(0 = y_{initial} + v_{iy}t - \frac{1}{2}gt^2\)

we can determine the car's flight duration and thus the horizontal distance traveled during this time. The shape of its path characterizes a parabola and is symmetrical around its peak. Understanding the principles that govern the parabolic trajectory allows us to predict where and when the car will impact the ocean's surface.