4.7:
Projectile Motion: Example
Consider an archer shooting an arrow such that it follows a projectile trajectory. Recall that the range of a projectile depends on the square of the initial speed and the sin2θ.
Now, sin2θ has a maximum value of 1, when theta equals 45°. In this case, the range of the projectile would be maximum for the given initial velocity.
Consider two arrows launched at angles 30 and 60°, having the same initial velocity of 50 meters per second. The acceleration due to gravity for both the arrows is 9.8 meters per second squared.
Therefore, substituting the velocity and angle values, the range covered by both the arrows equals 220.9 meters.
Here, since sin(180 − sin2θ) is equal to 2θ, the range of the projectile for complementary launch angles is the same.
However, the maximum height each arrow attains is proportional to the square of the initial velocity and sin2θ. Therefore, the maximum height is different for both cases.
4.7:
Projectile Motion: Example
The theory of projectile motion is very useful for players of several sports to improve their performance. For example, a javelin thrower needs to throw their javelin in such a way that it travels as far as possible. The javelin thrower takes a short run-up to increase the initial speed of the javelin. The range of a projectile is at its maximum at a 45° angle so javelin throwers try to angle their throw as close to 45° as possible.
When we speak of the range (R) of a projectile on level ground, we assume R is very small compared with the circumference of the Earth. If, however, the range is large, the Earth curves away below the projectile, and the acceleration resulting from gravity changes direction along the path. The range is larger than predicted by the range equation given for level ground because the projectile has farther to fall.
If the initial speed is large enough, the projectile goes into orbit. The Earth's surface drops 5 m every 8000 m. In 1 s, an object falls 5 m without air resistance. Thus, if an object has a horizontal velocity of 8000 m/s near the Earth's surface, it will go into orbit around the planet because the surface continuously falls away from the object. This is roughly the speed of a space shuttle in a low Earth orbit (when they were operational) or any satellite in a low Earth orbit.
This text is adapted from Openstax, University Physics Volume 1, Section 4.3: Projectile Motion.