You release a ball from rest at the top of a ramp and decide to assume that the ball’s acceleration is constant as it speeds up moving down the ramp. You observe the ball at three times while it is on the ramp: \(t_A = 1.5 \mathrm{~s}\text{,}\)\(t_B = 3.0 \mathrm{~s}\text{,}\) and \(t_C = 4.5 \mathrm{~s}\text{.}\) Using these observations, you determine \(\Delta v_{AB}\) (the change in velocity between \(t_A\) and \(t_B\)) and \(\Delta v_{BC}\) (the change in velocity between \(t_B\) and \(t_C\)).
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?
