Problem 30 - Calculus Early Transcendentals: Pearson New International Edition

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?

Step-by-Step Solution

Answer

The initial velocity of the block was 5 ft/s.

Step 1: Identify Known Variables

We are given that the block's acceleration, \(a\), is \(8 \, \text{ft/s}^2\), the length of the inclined plane, \(s\), is \(75\) feet, and the time, \(t\), it took to reach the bottom is \(3.75\) seconds. We need to find the initial velocity, \(v_0\).

Step 2: Use Kinematic Equation

We can use the kinematic equation that relates distance, initial velocity, acceleration, and time: \[s = v_0 t + \frac{1}{2} a t^2\]Substitute the known values: \[75 = v_0 \times 3.75 + \frac{1}{2} \times 8 \times (3.75)^2\]

Step 3: Simplify and Solve for Initial Velocity

Calculate the term \(\frac{1}{2} \times 8 \times (3.75)^2\):- \(\frac{1}{2} \times 8 = 4\)- \((3.75)^2 = 14.0625\)- \(4 \times 14.0625 = 56.25\)Now substitute back into the equation:\[75 = v_0 \times 3.75 + 56.25\]Rearrange to solve for \(v_0\):\[75 - 56.25 = v_0 \times 3.75\]\[18.75 = v_0 \times 3.75\]Divide both sides by 3.75:\[v_0 = \frac{18.75}{3.75} = 5 \, \text{ft/s}\]

Step 4: Verify the Calculation

Re-check the substitution and calculations:- Calculated \(56.25\) for the part involving acceleration correctly.- Correctly rearranged to \(v_0 = \frac{18.75}{3.75} = 5 \, \text{ft/s}\).The solution steps confirm that \(v_0 = 5 \, \text{ft/s}\) is correct.

Key Concepts

Understanding Initial Velocity

The initial velocity, often denoted as \( v_0 \), is the speed at which an object starts its motion before any forces such as acceleration have affected its velocity. In our exercise, we needed to find the block's initial velocity as it begins sliding down the inclined plane. Think of initial velocity as the object's starting speed. If the object was at rest and began moving, affecting factors like gravity and the angle of the inclined plane would change its state from rest to motion. In problems like these:

• We often need other known values like time, distance, and acceleration to compute initial velocity.
• Initial velocity can be zero when an object starts from a complete stop or some value if it’s already moving.

Determining initial velocity is key in understanding an object's motion over time.

Importance of Constant Acceleration

Constant acceleration means the rate of change of velocity is steady over time. In the given exercise, the block slides down an inclined plane with a constant acceleration of \( 8 \, \text{ft/s}^2 \). This simple, steady accelerating motion makes it easier to predict how the block's speed and position change over time. Key points about constant acceleration:

• The velocity of the object increases linearly with time if acceleration is constant.
• It allows us to use kinematic equations to calculate and predict various parameters of motion, such as distance, velocity, and time.

Understanding constant acceleration helps in solving problems where forces acting on an object remain unchanged, such as gravity on a uniform plane.

Kinematic Equations in Motion Analysis

Kinematic equations are invaluable tools for understanding motion in a straight line under uniform acceleration. These equations relate the five key parameters of motion: initial velocity \( v_0 \), final velocity \( v \), acceleration \( a \), time \( t \), and displacement \( s \). For our exercise, the relevant kinematic equation was:\[ s = v_0 t + \frac{1}{2} a t^2 \]This equation helped us find the block's initial velocity by using the given values of distance, time, and acceleration. Key takeaways about kinematic equations:

• They model the complexities of movement in a simplified mathematical form.
• Essential for solving real-world problems involving moving objects under uniform acceleration.
• Offer a systematic way to connect different aspects of motion.

With these equations, analyzing situations where objects fall, are thrown, or slide becomes much more manageable and accurate.