Question

1. a plane accelerates down a runway at 3 m/s² for 10 seconds before takeoff. what is the final velocity of the plane? v = 0.3 v = at 14. a motorcycle moving at 25 m/s slows down to 10 m/s in 3 seconds. what is the acceleration? a = -5 15. a car is initially at 10 m/s. it accelerates at 2 m/s² for 6 seconds. what is its final velocity? 16. an object is dropped from a height and reaches 24 m/s after 3 seconds. what is the objects average acceleration? 17. a soccer ball is kicked and accelerates at 5 m/s² for 0.6 seconds. if it started at 2 m/s, what is its final speed? 18. a rocket is launched upward. it starts from rest and accelerates at 9 m/s² for 12 seconds. what is its final velocity? 19. a skateboarder slows from 8 m/s to a stop in 4 seconds. what is the acceleration? 20. a cyclist is riding at 10 m/s when a pedestrian steps into the bike lane 15 meters ahead. the cyclist brakes with an acceleration of -2.5 m/s². will the cyclist stop in time, or hit the pedestrian? (assume instant reaction time and constant deceleration. show calculations and explain.) 21. a train is moving at 25 m/s when the conductor sees a barrier 200 meters ahead. the train begins braking immediately with a constant acceleration of -1.2 m/s². will the train stop before hitting the barrier? (show your work and explain your reasoning.) 22. a motorcycle is cruising at 18 m/s when the rider sees a red light 40 meters away. the rider brakes with a constant acceleration of -6 m/s². does the motorcycle stop before the intersection, or does it run the red light? (include all work and a short explanation.)

Explanation:

Step1: Recall the kinematic - equation

The main kinematic equations used are $v = v_0+at$ and $v^{2}-v_{0}^{2}=2ax$.

Step15: Solve for the car's final velocity

Given $v_0 = 10\ m/s$, $a = 2\ m/s^{2}$, $t = 6\ s$. Using $v=v_0 + at$, we substitute the values:
$v=10+2\times6$
$v = 10 + 12$
$v=22\ m/s$

Step16: Solve for the object's average acceleration

Given $v_0 = 0\ m/s$ (dropped), $v = 24\ m/s$, $t = 3\ s$. Using $a=\frac{v - v_0}{t}$, we substitute the values:
$a=\frac{24 - 0}{3}$
$a = 8\ m/s^{2}$

Step17: Solve for the soccer - ball's final speed

Given $v_0 = 2\ m/s$, $a = 5\ m/s^{2}$, $t = 0.6\ s$. Using $v=v_0+at$, we substitute the values:
$v=2+5\times0.6$
$v=2 + 3$
$v = 5\ m/s$

Step18: Solve for the rocket's final velocity

Given $v_0 = 0\ m/s$ (starts from rest), $a = 9\ m/s^{2}$, $t = 12\ s$. Using $v=v_0+at$, we substitute the values:
$v=0+9\times12$
$v = 108\ m/s$

Step19: Solve for the skate - boarder's acceleration

Given $v_0 = 8\ m/s$, $v = 0\ m/s$, $t = 4\ s$. Using $a=\frac{v - v_0}{t}$, we substitute the values:
$a=\frac{0 - 8}{4}$
$a=-2\ m/s^{2}$

Step20: Determine if the cyclist hits the pedestrian

First, use $v^{2}-v_{0}^{2}=2ax$ to find the stopping distance $x$. Given $v_0 = 10\ m/s$, $v = 0\ m/s$, $a=-2.5\ m/s^{2}$.
$0-(10)^{2}=2\times(-2.5)x$
$- 100=-5x$
$x = 20\ m$. Since $20\ m>15\ m$, the cyclist will hit the pedestrian.

Step21: Determine if the train stops before hitting the barrier

Use $v^{2}-v_{0}^{2}=2ax$. Given $v_0 = 25\ m/s$, $v = 0\ m/s$, $a=-1.2\ m/s^{2}$.
$0-(25)^{2}=2\times(-1.2)x$
$-625=-2.4x$
$x=\frac{625}{2.4}\approx260.42\ m$. Since $260.42\ m>200\ m$, the train will hit the barrier.

Step22: Determine if the motorcycle stops before the intersection

Use $v^{2}-v_{0}^{2}=2ax$. Given $v_0 = 18\ m/s$, $v = 0\ m/s$, $a=-6\ m/s^{2}$.
$0-(18)^{2}=2\times(-6)x$
$-324=-12x$
$x = 27\ m$. Since $27\ m>40\ m$, the motorcycle will stop before the intersection.

Answer:

1. $22\ m/s$
2. $8\ m/s^{2}$
3. $5\ m/s$
4. $108\ m/s$
5. $-2\ m/s^{2}$
6. The cyclist will hit the pedestrian.
7. The train will hit the barrier.
8. The motorcycle will stop before the intersection.