Answer to Question #306479 in Mechanics | Relativity for erick

Question

This exercise is included to exercise your ability to communicate about kinematics in 2

spatial dimensions. This classic class of kinematics problems is known as projectile motion.

A cannon ball is fired across a level field with an initial speed of 15m/s at an angle of 30◦ above

the horizontal. Given that the acceleration of the cannon ball is 9.8 m

s2 downward for the entire

flight of the ball,

questions to be answered:

1)construct a reference frame

2)construct a vector equation describing the coordinates of the cannon ball as a function of time

3) construct a system of equations describing the coordinates of the cannon ball as a function of time

4)eliminate time from that system of equations to show that the path of the cannon ball is

parabolic

5) calculate the time at which the ball strikes the ground (using your coordinate system)

6) calculate the horizontal distance covered by the ball before it strikes the ground. This is

called the range of the projectile.

Answer 1

Explanations & Calculations

1)

• Take the starting point as the (0,0)/origin of the coordinate system XY and any measure measured or a quantity directed upwards is considered positive and negative otherways.

2)

$\qquad\qquad \begin{aligned} \small \underline{r}&=\small x\hat i+y\hat j\\ &=\small (v\cos\theta t)\hat i+(v\sin\theta t-\frac{1}{2}gt^2)\hat j\\ &=\small (12.99t)\hat i+(7.5t-4.9t^2) \end{aligned}$

3)

$\qquad\qquad \begin{aligned} \small x&=\small v\cos\theta t \qquad\qquad y=v\sin\theta t-\frac{1}{2}gt^2\\ &=\small12.99t \qquad\qquad\,\,\,\,\,\,= 7.5t-4.9t^2\\ \end{aligned}$

4)

$\qquad\qquad \begin{aligned} \small t&=\small \frac{x}{12.99}\\ \small y&=\small 7.5\times\frac{x}{12.99}-4.9\times\frac{x^2}{168.74}\\ \small y&=\small -0.03x^2+0.58x\quad\equiv \quad y=ax^2+bx+c\\ \end{aligned}$ hence parabolic.

5)

$\qquad\qquad \begin{aligned} \small y&=\small 0\\ \small 0&=\small 7.5t-4.9t^2\\ \small t&=\small 0\,s\quad or\quad 1.5\,s\\ \small t&=\small 1.5\,s \end{aligned}$

6)

$\qquad\qquad \begin{aligned} \small y&=\small 0\\ \small 0 &=\small -0.03x^2+0.58x\\ \small x&=\small 0\quad or\quad 19.3\\ \small x&=\small range=19.3\,m \end{aligned}$