Question

A block slides down an inclined plane with a constant acceleration of 8 feet per second per second. If the inclined plane is 75 feet long and the block reaches the bottom in \(3.75\) seconds, what was the initial velocity of the block?

Short answer

The initial velocity of the block is 5 feet/s.

Step-by-step solution

Identify Known Variables

We know that the acceleration \(a\) is 8 feet/s², the distance \(s\) is 75 feet, and the time \(t\) is 3.75 seconds. We need to find the initial velocity \(v_0\).

Choose the Right Equation

Use the kinematic equation: \[ s = v_0 t + \frac{1}{2} a t^2 \]. This equation relates distance, initial velocity, time, and acceleration.

Substitute Known Values

Plug in the known values into the kinematic equation: \[ 75 = v_0 \times 3.75 + \frac{1}{2} \times 8 \times (3.75)^2 \].

Calculate the Acceleration Term

Calculate the value of the acceleration term: \( \frac{1}{2} \times 8 \times (3.75)^2 = 56.25 \).

Solve for Initial Velocity

Rearrange the equation to solve for \(v_0\): \[ v_0 = \frac{75 - 56.25}{3.75} \].

Perform the Calculation

Calculate: \( v_0 = \frac{18.75}{3.75} = 5 \). Thus, the initial velocity \(v_0\) is 5 feet/s.

Key concepts

Understanding Constant Acceleration

When an object moves with constant acceleration, it means that its velocity changes at a steady rate over time. This is a common scenario in many physics problems.
Constant acceleration can result from several forces acting on an object, gravity being the most typical one when dealing with inclined planes. In this exercise, the acceleration of the block is - constant, - 8 feet per second squared (8 feet/s²).
This means for every second the block slides down, its speed increases by 8 feet per second.
This concept is crucial as it helps predict how fast or how slow something will increase speed over time.
Kinematic equations rely heavily on knowing the type of acceleration to correctly model an object's motion. Using these equations, we can determine not only velocity but also distance and time considering this simple, linear acceleration.

Role of Initial Velocity

Initial velocity is the speed at which an object begins its motion. It's essential in determining how quickly an object reaches a destination.
In the described problem, the block starts its slide down the inclined plane. We initially do not know its starting speed, but we know it affects the total time and distance of the slide.
Initial velocity (- denoted as \(v_0\)), can sometimes be zero (if an object starts from rest), but in this case, we calculated it as 5 feet per second.
Determining this initial velocity is crucial as it also affects the object's final speed when combined with acceleration over time. Using the kinematic equation \( s = v_0 t + \frac{1}{2} a t^2 \), we can plug the values for time, acceleration, and distance to solve for \(v_0\). This insight allows us to understand the speed performance of objects over inclined planes or even flat surfaces.

Inclined Plane Dynamics

An inclined plane is a flat surface tilted at an angle, creating a scenario where gravity causes an object to accelerate down its slope.
Inclined planes modify the forces acting on an object, making them decrease the amount of force needed to move it.- On inclined planes, objects naturally accelerate downwards due to gravity, which is why they are used to demonstrate principles of mechanics.
In this exercise, the plane is specified as being 75 feet long. The slope length is crucial because it becomes the distance (\(s\)) the block travels.
Understanding how inclined planes work aids in grasping how gravity influences movement, altering how speed and distance are calculated. The principles remain straightforward but require careful attention to the plane's angle and length to accurately predict an object's motion.

Exploring the Distance-Time Relationship

The relationship between distance and time is fundamental in physics and is often represented with kinematic equations.
This problem uses the equation \( s = v_0 t + \frac{1}{2} a t^2 \) to relate these components with initial velocity and acceleration.
The distance covered by an object (\(s\)) is based on the combination of its initial velocity and the distance added from acceleration over a certain time (\(t\)).
For this problem, a block covers a distance of 75 feet during a total time of 3.75 seconds.

• This connection informs us of the object's travel pace and can be exploited to find unknowns like initial velocity.
• By knowing the exact factors like acceleration and time, we can predict not only how far but also how fast an object reaches a certain point.

Understanding this relationship empowers predictions and calculations related to real-world scenarios like vehicle movements, projectiles, and everyday objects influenced by gravity and other forces.