You are waiting in your car as a train moves northward at a speed of \(45 \mathrm{~mph}\text{.}\) As you wait, you notice two adventurers walking along the top of the train. Relative to you, Adventurer Jones is moving north at \(55 \mathrm{~mph}\text{,}\) while Adventurer Marks is moving north at \(30 \mathrm{~mph}\text{.}\) You then observe Adventurer Jones throw a golden idol toward Adventurer Marks; relative to you, the golden idol is moving south at \(20 \mathrm{~mph}\text{.}\) Determine the velocity of each relevant object (you, the train, Adventurer Jones, Adventurer Marks, and the golden idol) relative to the train.
At a certain moment in time, an object was located at \(\vec{r}_1 = \hat{x} + 4 \hat{y} \mathrm{~m}\text{.}\) At some later moment, it is located at \(\vec{r}_2 = -1 \hat{x} - 1 \hat{y} \mathrm{~m}\text{.}\) The object’s average velocity for this motion was \(\vec{v}_{ave} = -0.2 \hat{x} -0.5 \hat{y} \mathrm{~m/s}\text{.}\) How much time elapsed during the motion?
At one moment, an object was at location \(\vec{r}_1 = \hat{x} + 5 \hat{y} \mathrm{~m}\text{,}\) traveling with velocity \(\vec{v}_1 = 3 \hat{x} + 2 \hat{y} \mathrm{~m/s}\text{.}\) At some later moment, the object was traveling with velocity \(\vec{v}_2 = 5 \hat{x} + 7 \hat{y} \mathrm{~m/s}\text{.}\) The average acceleration of the object during this time period was \(\vec{a}_{ave} = 0.1 \hat{x} + 0.25 \hat{y} \mathrm{~m/s^2}\text{.}\) How much time elapsed during this motion?
An object, initially located at the origin, is traveling with velocity \(\vec{v}_i = -\hat{x} - 2 \hat{y} \mathrm{~m/s}\text{.}\) The object travels for \(4.0 \mathrm{~s}\) with an average acceleration of \(\vec{a}_{ave} = 0.25 \hat{x} + 0.5 \hat{y} \mathrm{~m/s^2}\text{.}\) At the end of the \(4.0 \mathrm{~s}\text{,}\) what is the velocity of the object?
A jumbo jet, flying northward, is landing with a speed of \(70 \mathrm{~m/s}\text{.}\) Once the jet touches down, it has \(800 \mathrm{~m}\) of straight, level runway in which to reduce its speed to \(5.0 \mathrm{~m/s}\text{.}\) Compute the \(x\)-component of the jet’s average acceleration during the landing. Assume north is the positive \(x\)-direction.
The driver of a sports car, traveling at \(10.0 \mathrm{~m/s}\) in the positive \(x\)-direction, steps down hard on the accelerator for \(5.0 \mathrm{~s}\text{.}\) As a result, the velocity increases to \(30.0 \mathrm{~m/s}\text{.}\) What was the average \(x\)-component of acceleration of the car during that \(5.0 \mathrm{~s}\) time interval?
