# Which projectile spends more time in the air the one fired from 30∘ or the one fired from 60∘?

Which projectile spends more time in the air, one fired at 30 degrees or one fired at 60 degrees?

## General guidance

Concepts and reason

The concepts required to solve the given problem are kinematic equations of motion. Initially, obtain the expression for time of flight for a projectile launched with an angle by using kinematic equation that relates initial speed, final speed, acceleration and time. Then, use the obtained expression for time of flight to determine the projection angle with more time of flight.

Fundamentals

The kinematic equations are a set of equations that are used to describe the motion of an object moving with constant acceleration. The kinematic equation that relates final velocity v, initial velocity , acceleration a and time is given by following expression. Assume a particle launched at angle with a velocity . The vertical component of velocity of the particle is given by following expression. The horizontal component of velocity of the particle is given by following expression. ## Step-by-step

### Step 1 of 2

Assume a projectile launch as shown in the below figure. Here, is the initial velocity, v is the velocity at maximum height and is the angle of projection. Use the following kinematic equation in vertical direction to find the time to reach maximum height. Here, is the vertical component of initial velocity, is the vertical acceleration, and is the time to reach maximum height. Substitute 0 for v, for  for in the above equation to solve for time. The total time of flight t of the projectile is equal to twice the time to reach maximum height  Substitute in the above equation  The vertical acceleration for the projectile after launch is equal to acceleration due to gravity g since only force acting on it is gravitational force.

### Step 2 of 2

The expression for time of flight is given as follows: From the above expression, the time of flight for projectile is directly proportional to sine value of angle of projection. Thus, for large angle time spent is large.

Therefore, projectile launched with spent more time in air.

The projectile fired with angle spent more time in air.

The value of sine angle increases with increase in angle. Thus, the sine value greater for angle than that of angle Therefore, time of flight is more for angle than for angle The projectile fired with angle spent more time in air.