Bullet Trajectories
Step 4: Using Bullet Trajectories to Calculate Distance and Height
Take your time as you proceed through this section because it is important that you understand the concepts involved in the calculations. There is a video below that will walk you through the process of finding the height of a shooter using some basic crime-scene measurements.
When you feel that you understand the concepts, gather your lab partners and complete the Bullet Trajectory Analysis Lab. Copies of the lab can be picked up from your teacher.
Calculating Angle of Impact
When gunshot holes are found at a crime scene, investigators often try to "reconstruct" the scene to better understand the events that transpired. One way to do this is by determining the angle in which the bullet hits an object, also knows as the impact angle. Today, criminal investigators use lasers or carefully-place dowels to determine angle of impact.
If dowels or lasers are not available, investigators can still calculate the angle of impact provided the bullet hole is "clean" enough. To accomplish this, the length and the width of the bullet hole is measured and trigonometry is used to calculate the angle.
Once the length and the width of the bullet hole are known, use this formula to calculate the impact angle (i):
Sin (i) = width / length
Using the measurements to the right we would plug them into the formual like this:
Sin (i) = 13 / 19
i = sin-1 (13 / 19)
i = 43o (rounding to the nearest degree)
Determining the Location of the Shooter
Now that the impact angle has been calculated, the location of the shooter can be determined by using the properties of right triangles. The three interior angles of a triangle must always add up to 180°. One of the angles will be 90°, representing the angle between the floor and the wall.
Therefore, if the second angle (the impact angle) is determined from the elliptical bullet hole left in the wall, the third angle can be calculated. Using the impact angle calculation above, angle A is 43° and angle B is 90°, which is the angle between the wall and the floor.
Since all three interior angles must add up to 180°, the unknown angle can be calculated: 180 – 90 – 43 = 47°. Angle C is known as the angle of elevation or angle of depression. If the bullet hole is higher than the shooter’s shoulder, it would be an angle of elevation; if lower, it is known as an angle of depression.
If the height of the the suspect is known, you should have enough information to determine the distance the shooter was from the target. In this case, the bullet hole is in the wall at 93″ above the floor, and the height of the suspect is 6′3″ (75″).
Assuming the suspect’s head is 8″ tall, the shoulder height of the suspect would be 5′7″ (67″). Since the bullet hole is above the shooter’s shoulder, the angle from the shooter would be considered an angle of elevation.
With this information, we can determine the distance the shooter was from the wall when the bullet was fired. The length of side a would be calculated by subtracting the height of the suspect’s shoulder from the height of the bullet: 93″ – 67″ = 26″. The length of segment a is 26″ and the angle of elevation is 43°. The tangent function can now be used to determine the length of b. Tan = Opposite/Adjacent. See the diagram below.
Determining the Height of a Shooter
Now that you know how to calculate the distance a shooter was from a target, you also know how to do the opposite: you can calculate the height of the shooter if you know the muzzle to target distance. One thing to keep in mind with these type of calculations is that we are assuming the bullet is exiting the gun at about the same height of the shooter's shoulder.
Let's work through a hypothetical situation together:
When you feel that you understand the concepts, gather your lab partners and complete the Bullet Trajectory Analysis Lab. Copies of the lab can be picked up from your teacher.
Once you have completed the lab, you are ready to help solve the crime!