$($a$)$

Imagine that a space probe could be fired as a projectile from the Earth's surface with an initial speed of $5\mathrm{.}80104$ m$/$s relative to the Sun. What would its speed be when it is very far from the Earth $($in m$/$s$)?$ Ignore atmospheric friction, the effects of other planets, and the rotation of the Earth. $($Consider the mass of the Sun in your calculations.$)$

Your response differs from the correct answer by more than $10\%.$ Double check your calculations. m$/$s

$($b$)$

What If$?$ The speed provided in part $($a$)$ is very difficult to achieve technologically. Often, Jupiter is used as a "gravitational slingshot" to increase the speed of a probe to the escape speed from the solar system, which is $1\mathrm{.}85104$ m$/$s from a point on Jupiter's orbit around the Sun $($if Jupiter is not nearby$).$ If the probe is launched from the Earth's surface at a speed of $4\mathrm{.}10104$ m$/$s relative to the Sun, what is the increase in speed needed from the gravitational slingshot at Jupiter for the space probe to escape the solar system $($in m$/$s$)?($Assume that the Earth and the point on Jupiter's orbit lie along the same radial line from the Sun.$)$

m$/$s

Imagine that a space probe could be fired as a projectile from the Earth's surface with an initial speed of $5\mathrm{.}80104$ m$/$s relative to the Sun. What would its speed be when it is very far from the Earth $($in m$/$s$)?$ Ignore atmospheric friction, the effects of other planets, and the rotation of the Earth. $($Consider the mass of the Sun in your calculations.$)$

Your response differs from the correct answer by more than $10\%.$ Double check your calculations. m$/$s

$($b$)$

What If$?$ The speed provided in part $($a$)$ is very difficult to achieve technologically. Often, Jupiter is used as a "gravitational slingshot" to increase the speed of a probe to the escape speed from the solar system, which is $1\mathrm{.}85104$ m$/$s from a point on Jupiter's orbit around the Sun $($if Jupiter is not nearby$).$ If the probe is launched from the Earth's surface at a speed of $4\mathrm{.}10104$ m$/$s relative to the Sun, what is the increase in speed needed from the gravitational slingshot at Jupiter for the space probe to escape the solar system $($in m$/$s$)?($Assume that the Earth and the point on Jupiter's orbit lie along the same radial line from the Sun.$)$

m$/$s

Imagine that a space probe could be fired as a projectile from the Earth's surface with an initial speed of $5\mathrm{.}80104$ m$/$s relative to the Sun. What would its speed be when it is very far from the Earth $($in m$/$s$)?$ Ignore atmospheric friction, the effects of other planets, and the rotation of the Earth. $($Consider the mass of the Sun in your calculations.$)$

Your response differs from the correct answer by more than $10\%.$ Double check your calculations. m$/$s

$($b$)$

What If$?$ The speed provided in part $($a$)$ is very difficult to achieve technologically. Often, Jupiter is used as a "gravitational slingshot" to increase the speed of a probe to the escape speed from the solar system, which is $1\mathrm{.}85104$ m$/$s from a point on Jupiter's orbit around the Sun $($if Jupiter is not nearby$).$ If the probe is launched from the Earth's surface at a speed of $4\mathrm{.}10104$ m$/$s relative to the Sun, what is the increase in speed needed from the gravitational slingshot at Jupiter for the space probe to escape the solar system $($in m$/$s$)?($Assume that the Earth and the point on Jupiter's orbit lie along the same radial line from the Sun.$)$

m$/$s