A quarterback is standing on the football field preparing to throw a pass. His receiver is
standing 20 yards down the field and 15 yards to the quarterback's left. The quarterback
throws the ball at a velocity of 60 mph towards the receiver at an upward angle of 30° (see
the following figure). Write the initial velocity vector of the ball V, in component form.
If you draw a diagram, you can see that the direction of the ball is at an angle θ left of directly downfield, where tanθ = 3/4
So, if we set up an x-y-z coordinate system with origin at the quarterback, and
downfield in the +x direction, then the components of v (in mi/hr) are
vx = 60 cos30° cosθ = 60 * √3/2 * 2/5 √3
vy = 60 cos30° sinθ = 60 * √3/2 * 3/5 = 18/5 √3
vz = 60sin30° = 30
To write the initial velocity vector of the ball V in component form, we need to break the velocity into its horizontal and vertical components.
Given:
- Velocity magnitude: 60 mph
- Angle of 30°
To find the horizontal component of velocity (Vx):
Vx = (Velocity magnitude) * cos(angle)
= 60 mph * cos(30°)
To find the vertical component of velocity (Vy):
Vy = (Velocity magnitude) * sin(angle)
= 60 mph * sin(30°)
Therefore, the initial velocity vector V can be written in component form as:
Vx = 60 mph * cos(30°)
Vy = 60 mph * sin(30°)
To write the initial velocity vector of the ball V in component form, we need to break it down into its horizontal and vertical components.
Given:
Initial velocity of the ball V = 60 mph
Upward angle of 30°
To find the horizontal component, we need to find the velocity of the ball in the horizontal direction. This can be found using the formula:
Horizontal component = Initial velocity * cos(angle)
In this case, the initial velocity is 60 mph and the angle is 30°. So, we can calculate the horizontal component as follows:
Horizontal component = 60 mph * cos(30°)
To find the vertical component, we need to find the velocity of the ball in the vertical direction. This can be found using the formula:
Vertical component = Initial velocity * sin(angle)
In this case, the initial velocity is 60 mph and the angle is 30°. So, we can calculate the vertical component as follows:
Vertical component = 60 mph * sin(30°)
Therefore, the initial velocity vector of the ball V in component form is:
V = (Horizontal component, Vertical component)
Now, let's calculate the values:
Horizontal component = 60 mph * cos(30°) = 60 * 0.866 = 51.96 mph
Vertical component = 60 mph * sin(30°) = 60 * 0.5 = 30 mph
So, the initial velocity vector V is:
V = (51.96 mph, 30 mph)