Questions: Given the velocity v=ds/dt and the initial position of a body moving along a coordinate line, find the body's position at time t. v=9.8 t+7, s(0)=17 s(t)=

Given the velocity v=ds/dt and the initial position of a body moving along a coordinate line, find the body's position at time t.
v=9.8 t+7, s(0)=17
s(t)=

Transcript text: Given the velocity $\mathrm{v}=\frac{\mathrm{ds}}{\mathrm{dt}}$ and the initial position of a body moving along a coordinate line, find the body's position at time $t$. \[ v=9.8 t+7, s(0)=17 \] $s(t)=$ $\square$

Solution

Solution Steps

To find the body's position at time $ t $, we need to integrate the velocity function $ v(t) $ with respect to time $ t $. The initial position $ s(0) = 17 $ will be used to determine the constant of integration.

Solution Approach

Integrate the velocity function $ v(t) = 9.8t + 7 $ to find the position function $ s(t) $.
Use the initial condition $ s(0) = 17 $ to solve for the constant of integration.
Substitute the constant back into the position function to get the final expression for $ s(t) $.

Step 1: Define the Velocity Function

The velocity of the body is given by the equation: \[ v(t) = 9.8t + 7 \]

Step 2: Integrate the Velocity Function

To find the position function $ s(t) $, we integrate the velocity function: \[ s(t) = \int v(t) \, dt = \int (9.8t + 7) \, dt = 4.9t^2 + 7t + C \] where $ C $ is the constant of integration.

Step 3: Apply the Initial Condition

We know the initial position $ s(0) = 17 $. Substituting $ t = 0 $ into the position function gives: \[ s(0) = 4.9(0)^2 + 7(0) + C = C \] Setting this equal to the initial position: \[ C = 17 \]

Step 4: Write the Final Position Function

Substituting $ C $ back into the position function, we have: \[ s(t) = 4.9t^2 + 7t + 17 \]