107–110. {Use of Tech} Motion with gravity Consider the following descriptions of the vertical motion of an object subject only to the acceleration due to gravity. Begin with the acceleration equation a(t) = v' (t) = -g , where g = 9.8 m/s² .
a. Find the velocity of the object for all relevant times. 
A payload is released at an elevation of 400 m from a hot-air balloon that is rising at a rate of 10 m/s.

Solve the initial-value problem: $x(x+1)\frac{dy}{dx}+xy=1$, $y\left(e\right)=1$. What is the solution $y\left(x\right)$?

Solve the initial value problem for the homogeneous differential equation $(x^2+2y^2)\frac{dx}{dy}=xy$, with the initial condition $y(-1)=2$. What is the explicit solution $y\left(x\right)$?

Solve the initial value problem using the method of Laplace transforms: $y^{\mathrm{\text{'}\text{'}}}+4y=0$, $y\left(0\right)=2$, $y^{\mathrm{\text{'}}}\left(0\right)=0$. What is $y\left(t\right)$?

Solve the initial value problem: $x^{\prime }=2x$, $x\left(0\right)=4$. What is $x\left(t\right)$?

107–110. {Use of Tech} Motion with gravity Consider the following descriptions of the vertical motion of an object subject only to the acceleration due to gravity. Begin with the acceleration equation a(t) = v' (t) = -g , where g = 9.8 m/s² .

d. Find the time when the object strikes the ground.

A payload is released at an elevation of 400 m from a hot-air balloon that is rising at a rate of 10 m/s.

107–110. {Use of Tech} Motion with gravity Consider the following descriptions of the vertical motion of an object subject only to the acceleration due to gravity. Begin with the acceleration equation a(t) = v' (t) = -g , where g = 9.8 m/s² .

b. Find the position of the object for all relevant times. 

A payload is released at an elevation of 400 m from a hot-air balloon that is rising at a rate of 10 m/s.

107–110. {Use of Tech} Motion with gravity Consider the following descriptions of the vertical motion of an object subject only to the acceleration due to gravity. Begin with the acceleration equation a(t) = v' (t) = -g , where g = 9.8 m/s² .

c. Find the time when the object reaches its highest point. What is the height? 

A payload is released at an elevation of 400 m from a hot-air balloon that is rising at a rate of 10 m/s.

Velocity to position Given the following velocity functions of an object moving along a line, find the position function with the given initial position.

v(t) = 2t + 4; s(0) = 0