A football is kicked at a velocity of 10.5 m/s, 45 degrees above the horizon. After the ball has traveled only horizontally 3.5 m, a member of the opposing team manages to tip the ball with a velocity of 4.5 m/s at an angle of 70 degrees above the horizon. What is the resultant vector of the ball after the tip?
You have just said what the vector (velocity) of the ball is just after the tip. (4.5 m/s at an angle of 70 degrees above the horizon)
What "resultant vector" are you talking about?
Before it is tipped, its vector velocity will have changed due to the action of gravity.
To find the resultant vector of the ball after the tip, we need to break down the initial velocity and the tip velocity into their horizontal and vertical components.
The initial velocity of the ball is 10.5 m/s at an angle of 45 degrees above the horizon. To find the horizontal and vertical components, we can use simple trigonometry.
Horizontal component of the initial velocity = initial velocity * cos(angle)
Vertical component of the initial velocity = initial velocity * sin(angle)
Horizontal component of the initial velocity = 10.5 m/s * cos(45 degrees) = 7.425 m/s
Vertical component of the initial velocity = 10.5 m/s * sin(45 degrees) = 7.425 m/s
Now, let's consider the tip velocity. The tip velocity is 4.5 m/s at an angle of 70 degrees above the horizon.
Horizontal component of the tip velocity = tip velocity * cos(angle)
Vertical component of the tip velocity = tip velocity * sin(angle)
Horizontal component of the tip velocity = 4.5 m/s * cos(70 degrees) = 1.484 m/s
Vertical component of the tip velocity = 4.5 m/s * sin(70 degrees) = 4.251 m/s
Next, we add the horizontal and vertical components separately to find the resultant horizontal and vertical components of the ball's velocity after the tip.
Resultant horizontal component = initial horizontal component + tip horizontal component
Resultant vertical component = initial vertical component + tip vertical component
Resultant horizontal component = 7.425 m/s + 1.484 m/s = 8.909 m/s
Resultant vertical component = 7.425 m/s + 4.251 m/s = 11.676 m/s
Finally, we can use these resultant horizontal and vertical components to find the resultant velocity and angle above the horizon.
Resultant velocity = square root of (Resultant horizontal component^2 + Resultant vertical component^2)
Resultant angle = arctan(Resultant vertical component / Resultant horizontal component)
Resultant velocity = square root of (8.909 m/s^2 + 11.676 m/s^2) = 14.83 m/s
Resultant angle = arctan(11.676 m/s / 8.909 m/s) = 1.192 radians or 68.42 degrees above the horizon
Therefore, the resultant vector of the ball after the tip is 14.83 m/s at an angle of 68.42 degrees above the horizon.