Vector methods with projectiles

​​In a nutshell

The use of vector methods makes it easier to solve projectile problems. You only have to take care of the signs of the vectors and remember to group them according to their corresponding unit vectors, $$\bold i$$ and $$\bold j$$.​

Equations

Variable definitions

Vector methods with projectiles

​​procedure

Example 1

A particle $$P$$ in the $$x\text-y$$ plane is projected from the origin. Its velocity after $$3.5$$ seconds is $$(5\bold i + 5 \bold j) \ ms^{-1}$$.

i) Find the initial velocity. 

ii) Find the initial speed to two decimal places.

iii) Find the angle the direction of motion makes with the positive $$x$$-axis at $$t=0$$.

i) Find the initial velocity. 

Using the expression $$\bold v = \bold u + \bold a t$$ you have that $$\bold u= \bold v - \bold a t$$. As you know that $$\bold a = (0 \bold i - 9.8 \bold j )\ ms^{-2}$$​: 

$$\bold u = \bold v - \bold a t = (5\bold i + 5 \bold j) - (0 \bold i - 9.8 \bold j )t = (5-0t)\bold i + ( 5+9.8t)\bold j$$​​

Substitute $$t=3.5 \ s$$:

​$$\bold u = 5\bold i + ( 5 + 9.8 \times 3.5)\bold j = \underline{(5 \bold i +39.3 \bold j) \ ms^{-1}}$$​​

ii) Find the initial speed. 

Use the formula $$Speed = \sqrt{v_x^2 + v_y^2}$$. Therefore:

$$Speed = \sqrt{v_x^2 + v_y^2} = \sqrt{(5)^2 + (39.3)^2} = \underline{39.61 \ ms^{-1} \ (2 \ d.p.)}$$​​

iii) Find the angle the direction of motion makes with the positive x-axis at $$t=0$$.

​​

​	Draw the vector of the velocity and resolve it into its horizontal and vertical components. Using basic trigonometry you have that:  ​$$\tan \alpha = \cfrac{39.3}{5}$$​​

Therefore, $$\underline{\alpha = 82.57 ^{\circ}}.$$​