Question

An \(8.5\) -kg object passes through the origin with a velocity of 42 \(\mathrm{m} / \mathrm{s}\) parallel to the \(x\) axis. It experiences a constant \(19-\mathrm{N}\) force in the direction of the positive \(y\) axis. Calculate \((a)\) the velocity and \((b)\) the position of the particle after \(15 \mathrm{~s}\) have elapsed.

Short answer

The velocity of the object after 15 seconds is \(42 \, m/s\) in the x-direction and \(33.53 \, m/s\) in the y-direction. The position of the object after 15 seconds is \(630 \, m\) in the x-direction and \(500.6 \, m\) in the y-direction.

Step-by-step solution

Identify given quantities and determine acceleration

The object has a mass of \(8.5 \, kg\) and initially moves with a velocity of \(42 \, m/s\) along the x-axis. A force of \(19 \, N\) is acting on it in the positive y-direction. According to Newton's second law, the acceleration of the object can be calculated using the equation \(a = F/m\), where \(F\) is the force and \(m\) is the mass of the object. Substituting the given values, we get \(a = 19 \, N/ 8.5 \, kg = 2.235 \, m/s²\) in the positive y-direction.

Calculate velocity after 15 seconds

The object's velocity can be calculated using the equation \(v = u + at\), where \(u\) is the initial velocity, \(a\) is the acceleration, and \(t\) is the time. Here, as the force is acting perpendicular to the motion of the object, the x-component of velocity remains unchanged, i.e., \(42 \, m/s\). The y-component of the velocity can be computed as \(v = 0 + (2.235 \, m/s²)(15 \, s) = 33.53 \, m/s\). Therefore, after 15 seconds, the object's velocity is \(42 \, m/s\) in the x-direction and \(33.53 \, m/s\) in the y-direction.

Calculate the position after 15 seconds

The object's position can be calculated using the equation \(s = ut + 1/2at²\), where \(s\) is the distance, \(u\) is the initial velocity, \(a\) is the acceleration, and \(t\) is the time. The x-component of the displacement remains unchanged, i.e., \(s_x = (42 \, m/s)(15 \, s) = 630 \, m\). The y-component of the displacement can be computed as \(s_y = 0 + 1/2(2.235 \, m/s²)(15 \, s)² = 500.6 \, m\). Therefore, after 15 seconds, the object's position is \(630 \, m\) in the x-direction and \(500.6 \, m\) in the y-direction.

Key concepts

Newton's Second Law

Newton's second law of motion is a fundamental principle that describes the relationship between the force applied to an object, its mass, and the resulting acceleration. The law is often expressed as the equation
\[\begin{equation} F = ma, \end{equation}\]
where \(F\) stands for force, \(m\) is the mass of the object, and \(a\) is the acceleration.This law implies that if you know two of the quantities (force, mass, acceleration), you can calculate the third. It's crucial in predicting how an object will move when forces are applied. Thus, whether you're calculating the thrust needed to launch a rocket or determining the force a car exerts to accelerate, Newton's second law provides the computational foundation. It's also the basis for many phenomena in classical mechanics, from the simple motion of a sliding block to the complex orbits of planets.

Acceleration Calculation

To calculate acceleration, we apply the formula derived from Newton's second law:
\[\begin{equation} a = \frac{F}{m}. \end{equation}\]
Acceleration is the rate at which the velocity of an object changes with time. In our example, an object receives a force in the y-direction which means its acceleration is only in the y-direction. There's no additional force in the x-direction, so there's no acceleration there—the horizontal velocity remains constant. The object's vertical velocity, however, will change due to the vertical force. Calculating acceleration requires a clear understanding of the direction and magnitude of forces, and how they affect an object's motion. This understanding underpins everything from engineering applications to everyday actions like braking a car or throwing a ball.

Velocity and Position after Time Elapsed

Velocity and position are key components of an object's motion, and their calculation after some time has elapsed is essential in many areas of physics.To find the velocity of an object after a certain time period, we combine the initial velocity with the product of acceleration and time, using the formula:
\[\begin{equation}v = u + at, \end{equation}\]
where \(v\) is the final velocity, \(u\) is the initial velocity, \(a\) is the acceleration, and \(t\) is the time elapsed.

Position Calculation

The position, on the other hand, depends on both the initial velocity and the object's acceleration over time. The equation used is:
\[\begin{equation} s = ut + \frac{1}{2}at^2, \end{equation}\]
where \(s\) is the displacement (position relative to the starting point). These calculations allow us to predict where an object will be and how fast it will be traveling at any point in its journey, which is fundamental for everything from navigation to planning space missions.