Find the position and velocity of an object moving along a straight line with the given acceleration, initial velocity, and initial position.

The velocity is
(Type an exact answer.)
The position is
(Type an exact answer.)

Ask by Zimmerman Medina. in the United States

Jan 23,2025

Question

Find the position and velocity of an object moving along a straight line with the given acceleration, initial velocity, and initial position.

The velocity is
(Type an exact answer.)
The position is
(Type an exact answer.)

Ask by Zimmerman Medina. in the United States

Jan 23,2025

UpStudy AI · Accepted Answer

To find the position $$ s(t) $$ of the object, we'll integrate the velocity function $$ v(t) $$ with respect to time, and then apply the initial condition $$ s(0) = 5 $$.

Given:
$$
v(t) = 7 + \frac{\sin(\pi t)}{\pi}
$$

**Step 1: Integrate the Velocity to Find Position**

$$
s(t) = \int v(t) \, dt = \int \left(7 + \frac{\sin(\pi t)}{\pi}\right) dt
$$

$$
s(t) = 7t - \frac{\cos(\pi t)}{\pi^2} + C
$$

**Step 2: Apply the Initial Condition**

Using $$ s(0) = 5 $$:

$$
5 = 7 \cdot 0 - \frac{\cos(0)}{\pi^2} + C
$$

$$
5 = -\frac{1}{\pi^2} + C \implies C = 5 + \frac{1}{\pi^2}
$$

**Step 3: Write the Final Expression for Position**

$$
s(t) = 7t - \frac{\cos(\pi t)}{\pi^2} + 5 + \frac{1}{\pi^2}
$$

$$
s(t) = 5 + 7t + \frac{1 - \cos(\pi t)}{\pi^2}
$$

**Final Answer:**
$$
s(t) = 5 + 7\, t + \dfrac{1 - \cos(\pi t)}{\pi^{2}}
$$
The position of the object is $$ s(t) = 5 + 7t + \frac{1 - \cos(\pi t)}{\pi^{2}} $$.

Find the position and velocity of an object moving along a straight line with the given acceleration, initial velocity, and initial position.

The velocity is
(Type an exact answer.)
The position is
(Type an exact answer.)

Upstudy AI Solution

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Find the position and velocity of an object moving along a straight line with the given acceleration, initial velocity, and initial position. The velocity is (Type an exact answer.) The position is (Type an exact answer.)

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