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## Class 11 Physics (India)

### Unit 8: Lesson 6

Projectiles launched at an angle

# Projectiles launched at an angle review

Learn about projectile motion vectors and how the launch angle impacts the trajectory.

## Key terms

TermMeaning
Launch angleThe angle of a projectile’s initial velocity when measured from the horizontal direction. These angles are typically 90, degree or less.

## Vectors of projectiles launched at an angle

### Constant vertical acceleration

The only acceleration of a projectile is the downwards acceleration due to gravity (see Figure 1 below). Vertical acceleration is always equal to 9, point, 8, start fraction, start text, m, end text, divided by, start text, s, end text, squared, end fraction downward at all points of the trajectory, no matter how a projectile is launched.
In the center is a horizontal black line. An arrow arcs rightward as a light blue parabola from one end of the parabola to the other. There are 5 small dark blue dots evenly spaced along the arc. Each dot has an identical purple arrow pointing downwards. The arrows are labeled "a = -g".
Figure 1. Acceleration a is constant, downwards, and equal to g

### No horizontal acceleration

Nothing accelerates a projectile horizontally, so horizontal acceleration is always zero.

### Horizontal velocity is constant

The projectile’s horizontal speed is constant throughout the entire trajectory (see figure 2 below) because gravity only acts downwards in the vertical direction.
In the center is a horizontal black line. An arrow arcs rightward as a light blue parabola from one end of the parabola to the other. There are 5 small dark blue dots evenly spaced along the arc. Each dot has an identical purple arrow pointing rightwards. The arrows are labeled "v_x".
Figure 2. Horizontal velocity is constant

### Vertical velocity changes direction and magnitude during trajectory

Before the object reaches the maximum height, the vertical speed v, start subscript, y, end subscript of a projectile decreases, because acceleration is in the opposite direction. The direction of the velocity is initially upward, since the object’s height is increasing (see Figure 3 below).
Vertical velocity becomes zero at the projectile’s maximum height. The vertical speed increases after the maximum height because acceleration is in the same direction (see figure 3 below). The direction of vertical velocity is downward as the object’s height decreases
In the center is a horizontal black line. An arrow arcs rightward as a light blue parabola from one end of the parabola to the other. There are 5 small dark blue dots evenly spaced along the arc. Each dot has purple vertical arrow. On the left side of the arc, the purple arrows point up and reduce to zero length at the top. On the right side of the arc, the purple arrows point down and increase towards the right.
Figure 3. A projectile’s constant downward acceleration a changes the vertical velocity v, start subscript, y, end subscript throughout the trajectory

## Analyzing angled launch trajectories

### Components of initial velocity

To see how to break down the total velocity vector into the horizontal and vertical components using trigonometry, see the article on analyzing vectors.
There are three right triangles. Each triangle's hypotenuse is a thick purple arrow, while its two legs are dashed purple lines. At the lower left vertex of each triangle is a blue dot. The hypotenuse of the first triangle is labeled "v sub o", its horizontal leg is labeled "v sub o x" and its vertical leg is labeled "v sub o y".
All three triangles' hypotenuses are the same length. The first triangle has its left vertex at a 45 degree angle, the second triangle at a 30 degree angle, and the third triangle at a 60 degree angle. As the angle changes, the lengths of the triangles' horizontal and vertical legs change too.
Figure 4. Changing the launch angles changes the initial velocity components

### Launch angle trajectory comparisons

The diagram below shows trajectories for different launch angles that have the same initial speed. The launch angle determines the maximum height, time in the air, and maximum horizontal distance of the projectile.
Along the images bottom is a horizontal black line. At the left of this black line is a cannon. From the barrel of the cannon, three dashed arrows shaped as parabolas originate and arc to the right of the black line. One purple parabola starts at 60 degree angle and is tall and terminates at 3/4 the way across the black line. Another purple parabola starts at 30 degree angle and is short and terminates at 3/4 the way across the black line also. A red parabola starts at 45 degree angle and is tall and terminates at the end of the black line.
Figure 5: Trajectories of launch angles with the same initial speed

### Higher launch angles have higher maximum height

The maximum height is determined by the initial vertical velocity. Since steeper launch angles have a larger vertical velocity component, increasing the launch angle increases the maximum height. (see figure 5 above).

### Higher launch angles have greater times in the air

The time in air is determined by the initial vertical velocity. Since steeper launch angles have a larger vertical velocity component, increasing the launch angle increases the time in air. For deeper explanations of the relationship between projectile time in air and initial vertical velocity, see Sal’s video on the optimal angle for a projectile.

### Projectile maximum horizontal distance depends on horizontal velocity and time in air

Launch angles closer to 45, degree give longer maximum horizontal distance (range) if initial speed is the same (see figure 5 above). These launches have a better balance of the initial velocity components that optimize the horizontal velocity and time in air (see figure 4).

## Common misconceptions

• People mix up horizontal vs. vertical components of acceleration and velocity. The acceleration is a constant downwards 9, point, 8, start fraction, start text, m, end text, divided by, s, squared, end fraction (see figure 1) because gravity is the only source of acceleration. This acceleration only changes the vertical velocity, so the horizontal velocity is constant.
• People can’t remember what is zero at the maximum height. Vertical velocity is zero at this point, but there is still horizontal velocity and acceleration is still down.