Problem 17 - Fundamentals of Physics

Question

A cart is propelled over an \(x y\) plane with acceleration components \(a_{x}=4.0 \mathrm{~m} / \mathrm{s}^{2}\) and \(a_{y}=-2.0 \mathrm{~m} / \mathrm{s}^{2} .\) Its initial velocity has components \(v_{0 x}=8.0 \mathrm{~m} / \mathrm{s}\) and \(v_{0 y}=12 \mathrm{~m} / \mathrm{s}\). In unit-vector notation, what is the velocity of the cart when it reaches its greatest \(y\) coordinate? ..18 A moderate wind accelerates a pebble over a horizontal \(x y\) plane with a constant acceleration \(\vec{a}=\left(5.00 \mathrm{~m} / \mathrm{s}^{2}\right) \hat{\mathrm{i}}+\left(7.00 \mathrm{~m} / \mathrm{s}^{2}\right) \hat{\mathrm{j}}\).

Step-by-Step Solution

Answer

The velocity is \( 32 \hat{i} \text{ m/s} \) when the cart is at its highest point.

Step 1: Identify the Key Concept

The cart's greatest y coordinate is reached when the vertical component of its velocity, \(v_y\), becomes zero. This is because the cart will momentarily stop moving upwards before starting to move downwards.

Step 2: Apply the Velocity Equation

The equation for final velocity with constant acceleration is given by \( v_y = v_{0y} + a_y t \). We want to find the time \( t \) when \( v_y = 0 \).

Step 3: Solve for Time

Set \( v_y = 0 \) and substitute the known values into the equation: \[ 0 = 12 + (-2)t \]Solve for \( t \): \[ 2t = 12 \] \[ t = 6 \text{ seconds} \].

Step 4: Calculate the x-component of Velocity

Using the equation \( v_x = v_{0x} + a_x t \), substitute \( t = 6 \) seconds:\[ v_x = 8 + 4 \times 6 \]\[ v_x = 8 + 24 = 32 \text{ m/s} \].

Step 5: Express the Velocity in Unit-Vector Notation

Since we know \( v_x = 32 \text{ m/s} \) and \( v_y = 0 \text{ m/s} \) at the greatest y coordinate, the velocity in unit-vector notation is:\( \vec{v} = 32 \hat{i} + 0 \hat{j} \text{ m/s} \).

Key Concepts

Acceleration Components

Acceleration components are crucial in understanding how an object moves, especially in projectile motion. They dictate how velocity changes over time in different directions. In our scenario, the cart has acceleration components:

• For the x-direction: \( a_x = 4.0 \text{ m/s}^2 \)
• For the y-direction: \( a_y = -2.0 \text{ m/s}^2 \)

These values tell us that the cart speeds up horizontally at \(4.0 \text{ m/s}^2\) and slows down vertically at \(2.0 \text{ m/s}^2\). The positive \(a_x\) means that the cart is accelerating to the right, while the negative \(a_y\) indicates deceleration upwards or acceleration downwards. This is integral in solving time-dependent problems.

Velocity Equation

The velocity equation helps us calculate how an object's velocity changes over time due to acceleration. For a body experiencing constant acceleration, the velocity can be described by the equation:\[ v = v_0 + at \]Here, \(v\) is the final velocity, \(v_0\) is the initial velocity, \(a\) is the constant acceleration, and \(t\) is time.

Finding the Greatest Y Coordinate

For our cart, the greatest y coordinate is reached when its vertical velocity component becomes zero. To find this, set \( v_y = 0 \) in the equation:\[ 0 = v_{0y} + a_y t \]Substituting the values, \(0 = 12 + (-2)t\), solve for time \(t\) to find the exact moment when this occurs.

Unit-Vector Notation

Expressing velocities in unit-vector notation clarifies vector directions in a two-dimensional plane. A unit vector has a magnitude of one and points in a standard direction, represented as \(\hat{i}\) for the x-axis and \(\hat{j}\) for the y-axis.In our example, when the cart reaches its peak y position, the y component of velocity is zero. Therefore, using unit-vector notation, the velocity vector \(\vec{v}\) is expressed as:\[ \vec{v} = 32 \hat{i} + 0 \hat{j} \text{ m/s} \]This makes it clear that only the x-component contributes to the velocity at that moment. It shows the horizontal movement of the cart precisely.

Final Velocity Calculation

Calculating the final velocity is an essential task in projectile motion problems. It involves using the known values of initial velocity, acceleration, and time. In our exercise, we calculated time as \(t = 6\) seconds at the peak y coordinate. Use this time to find the horizontal component of the final velocity using its initial velocity and acceleration:

Horizontal Velocity Calculation

\[ v_x = v_{0x} + a_x t \]\[ v_x = 8 + 4 \times 6 = 32 \text{ m/s} \]With this, we've determined that the cart's final velocity when reaching its greatest y position is 32 m/s in the horizontal direction. Understanding these calculations is crucial for mastering projectile motion.