Problem 43 - Thomas Calculus

Question

Give the acceleration \(a=d^{2} s / d t^{2}\) , initial velocity, and initial position of a body moving on a coordinate line. Find the body's position at time \(t\). \(a=-4 \sin 2 t, \quad v(0)=2, \quad s(0)=-3\)

Step-by-Step Solution

Answer

The body's position at time \(t\) is \(s(t) = \sin 2t - 3\).

Step 1: Integrate Acceleration to Find Velocity

Given the acceleration function \(a = -4 \sin 2t\), we find the velocity by integrating acceleration. The integral of \(-4 \sin 2t\) with respect to \(t\) is:\[ v(t) = \int -4 \sin 2t \, dt = 2\cos 2t + C_1 \]where \(C_1\) is a constant of integration. Since \(v(0) = 2\), we substitute \(t = 0\):\[ 2 = 2\cos 0 + C_1 \]\[ 2 = 2\cdot1 + C_1 \]\[ C_1 = 0 \]Thus, the velocity function is \(v(t) = 2\cos 2t\).

Step 2: Integrate Velocity to Find Position

Now that we have the velocity function \(v(t) = 2\cos 2t\), we find the position function by integrating velocity:\[ s(t) = \int 2\cos 2t \, dt \]\[ s(t) = \sin 2t + C_2 \]where \(C_2\) is a constant of integration. Given that \(s(0) = -3\), we substitute \(t = 0\):\[ -3 = \sin 0 + C_2 \]\[ -3 = 0 + C_2 \]\[ C_2 = -3 \]Thus, the position function is \(s(t) = \sin 2t - 3\).

Step 3: Conclusion

The position of the body at time \(t\) is given by the equation:\[ s(t) = \sin 2t - 3 \]

Key Concepts

Kinematics

Kinematics is the branch of physics that deals with the motion of objects without considering the forces that cause this motion. It focuses on various parameters such as displacement, velocity, and acceleration.

In our current problem, the body moves along a coordinate line, which is a fundamental scenario in kinematics, especially in one-dimensional motion. We are provided with an initial position, velocity, and acceleration.

Kinematic equations help us link these quantities to understand the object's behavior over time. By integrating known functions such as acceleration, we can determine velocity and eventually the position of a moving body. Remember:

• Acceleration is the rate of change of velocity.
• Velocity is the rate of change of position.
• Position is the exact point an object is at, at a particular time.

Understanding kinematics is crucial for solving problems in mechanics, whether simple or complex.

Acceleration

Acceleration is essentially how quickly an object's velocity changes. It is a vector quantity, which means it has both magnitude and direction.

In the context of this exercise, the given acceleration function is a sinusoidal function: \( a = -4 \sin 2t \). This signifies periodic changes in acceleration, influenced by the sine function:

• The negative sign indicates acceleration is in the opposite direction to the motion when the sine function is positive.
• The factor of 2 inside the sine function affects the frequency of oscillation.

To find the velocity from acceleration, integration is necessary. This integration process accounts for the constant change expressed by acceleration across time.

Velocity

Velocity refers to the speed and direction at which an object moves. Unlike speed, velocity is a vector and includes the direction of travel.

Here, we find velocity by integrating the given acceleration, \( a = -4 \sin 2t \).

• The integration yields the velocity function \( v(t) = 2\cos 2t + C_1 \).

The constant \( C_1 \) is determined using initial conditions, in this case \( v(0) = 2 \). Initial conditions are used to personalize the general solution to a specific situation.

Velocity is integral to predicting where an object will be at any given moment, as it tells us how the position changes over time.

Integration Techniques

Integration is a fundamental mathematical technique used to solve problems in physics, particularly in Newtonian mechanics. It helps find the function of a quantity when its rate of change is known.

In the exercise at hand, we use integration to find both velocity and position from our known function of acceleration.

First, integrating acceleration function \( a = -4 \sin 2t \) provides the velocity: \( v(t) = 2\cos 2t + C_1 \). By applying the initial velocity condition, we determine \( C_1 = 0 \).

Next, integrating the velocity function \( v(t) = 2\cos 2t \) gives us the position: \( s(t) = \sin 2t + C_2 \). Again, using the initial condition, the constant \( C_2 \) becomes \(-3 \).

Integration is thus indispensable in physics, allowing us to go from acceleration (a rate of change) back to a position, building a complete picture of motion from fundamental principles.