A car is parked on a steep incline overlooking the ocean,...

A car is parked on a steep incline overlooking the ocean, where the incline makes an angle of $37.0^{\circ}$ below the horizontal. The negligent driver leaves the car in neutral, and the parking brakes are defective. The car rolls from rest down the incline with a constant acceleration of $4.00 \mathrm{~m} / \mathrm{s}^{2}$, traveling $50.0 \mathrm{~m}$ to the edge of a vertical cliff. The cliff is $80.0 \mathrm{~m}$ above the ocean. Find (a) the speed of the car when it reaches the edge of the cliff and the time it takes to get there, (b) the velocity of the car when it lands in the ocean, (c) the total time the car is in motion, and (d) the position of the car when it lands in the ocean, relative to the base of the cliff.

A car is parked on a steep incline overlooking the ocean, where the incline makes an angle of $37.0^{\circ}$ below the horizontal. The negligent driver leaves the car in neutral, and the parking brakes are defective. The car rolls from rest down the incline with a constant acceleration of $4.00 \mathrm{~m} / \mathrm{s}^{2}$, traveling $50.0 \mathrm{~m}$ to the edge of a vertical cliff. The cliff is $80.0 \mathrm{~m}$ above the ocean. Find
(a) the speed of the car when it reaches the edge of the cliff and the time it takes to get there, (b) the velocity of the car when it lands in the ocean, (c) the total time the car is in motion, and (d) the position of the car when it lands in the ocean, relative to the base of the cliff.

Key Concepts

Constant Acceleration Kinematics

This concept involves the equations of motion that relate displacement, initial velocity, acceleration, and time when acceleration is constant. In scenarios such as a vehicle accelerating down an incline, these equations are used to calculate the speed after moving a certain distance and the total time of travel before the motion changes form (for example, from rolling to free flight).

Inclined Plane Dynamics

Inclined plane dynamics address the physics of motion along a slope, where the gravitational component acting along the incline causes acceleration. It involves analyzing forces and motion along an angled surface, which is crucial for understanding how a car accelerates down a hill before transitioning into projectile motion when leaving an elevated edge.

Projectile Motion

Projectile motion is the motion of an object that is launched into the air and is subject only to the acceleration due to gravity. Once the car leaves the edge of the cliff, its motion can be analyzed by decomposing it into horizontal and vertical components, allowing the use of kinematic equations to predict its trajectory, landing speed, and displacement relative to the base of the cliff.

Vector Decomposition

Vector decomposition involves breaking down a velocity or acceleration vector into its horizontal and vertical components. This is essential for solving problems where an object’s motion involves two dimensions, such as a car transitioning from accelerated motion on an incline to free-fall after leaving a cliff, enabling separate analyses of the horizontal and vertical motions.

Transcript

00:01 Hi, as for this problem, i think we are going to use the three equations related to constant acceleration motion.

00:12 Here i list in the right corner, and we can start to solve the problem step by step.

00:21 At first, part a, it has acceleration 4 meter per square second and distance traveled in this whole problem.

00:31 Was 50 meters and it has an initial velocity which is 0.

00:37 So we can write out this equation and plug all the numbers in.

00:47 0 times t which is 0 but i will leave it here for now 4 meter per square and t squared.

01:01 By solving this equation we can get the t is equal to 5 seconds and we all know v final equals v o plus a t so the velocity as the edge of the cliff was 5 seconds times times 4 meter per square second which is 20 meter per second now we can turn to our question b what is the velocity when it dropped on the ocean.

01:51 So in this process, we no longer have any force acting on the car because the engine is not working and we only have gravitational force taking effect.

02:09 And the vertical force can only affect vertical velocity.

02:15 So easier our step, we could get our final velocity in vertical and horizontal fractions.

02:32 So we divide the 20 meters into the 12 meter per second in vertical direction.

02:40 And the 16 meter per second in horizontal direction.

02:48 And we can write them out in case we forgot.

02:57 Horizontal one will be constant during the whole process, so it's 16 meter per second...

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