If you're seeing this message, it means we're having trouble loading external resources on our website.

If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked.

### Course: Physics archive>Unit 2

Lesson 2: Optimal angle for a projectile

# Optimal angle for a projectile part 1: Components of initial velocity

You want a projectile to fly as far as possible, at which angle should you launch it? We'll start with formulas for the initial velocity. Created by Sal Khan.

## Want to join the conversation?

• What do you do if the problem given does not have an angle? What equation do you use to find the angle?
• I think you can use reverse trigonometry, if it gives you two sides you can use Pythagorean because it'll be a right triangle, and you can plug in the side lengths for reverse trigonometry.
• What do scos and ssin mean?
• s is the velocity of the thrown object
scos is the horirontal component of the velocity while ssin is the vertical component of the velocity
• i noticed that you use the theta symbol and i would like to know what it means.
• It's just that: a symbol. Similar to x or y. It is Greek. It is commonly used to represent an angle.
• How would the problem change if you are given an initial or final height of something other than 0?
• how do you derive the maximum height reached by the object?
• I'm assuming you want the answer, and not a mathematical proof.
For just a quick review, the three most important equations in projectile motion are:
Δx = vt + (1/2)at^2
vfinal = vinitial+ at
(vfinal)^2 = (vinitial)^2 + 2aΔx

The maximum height of a projectile can be found from the formula (v)^2 = 2aΔy, where v is the initial vertical velocity of the projectile and a is the acceleration (most often 9.8 m/s/s, or "little g"). The formula can be rearranged to find the vertical displacement (maximum height):
Δy = v/2a
• Why are we using speed instead of velocity again?
• how do you write maximum height?
• why wouldn't it be velocity if it was a speed with a direction?
• When teacher want talk only about magnitude of velocity he use word "speed". Additional information, If your solving 1 dimensional vector problems including velocity you can keep the arrows at the top off they are not needed your sign take care of direction .
• can you find an angle if only given the the total time and total distance?
• Well first break up the initial velocity into components:
vx = vi cos θ (velocity in x direction)
Solve for vi:
vi = vx/cos θ
We know distance and time, so vx = d/t which is a value.
So vi = (d/t) / cos θ
vy = vi sin θ (initial velocity in y direction)
Substitute vi:
vy = [(d/t) / cos θ] sin θ
a = -9.8 m/s^2 is assumed
So we have initial velocity in y direction, acceleration, and time.
We can use the kinematic equation: vf = vy + a*t. If we substitute in what we know there are still 2 unknowns, vf and θ. What to do? We need another equation...
I know! We should halve the time. This is when we know vf = 0 m/s as the object is at its maximum height: