MATERIALS
• Football • Stopwatch
• Tape measure or a football field
• Calculator • Pencil and paper
• Willing friend
kick you can !
2 . When the ball hits the ground , stop the timer and mark where the ball first lands . Measure how far away that is from where you kicked it . A football field makes this pretty easy , but a long tape measure works just as well .
3 . Now for a bit of math . Divide the distance travelled ( d , meters ) by the amount of hang time ( t , seconds ). This tells you the ball ’ s horizontal speed , v x
, in meters per second : v x
= d t . 4 . Calculate the ball ’ s vertical speed , v y
, by multiplying half the hang time ( t ) by the acceleration due to gravity ( g = 9.8 m s 2 ): v x
2
+ v y2 = 1
2 gt
5 . Now it ’ s time to combine the horizontal and vertical speeds to get the total speed , v = √ ( v x2
+ v y2
). That ’ s how fast you kicked the ball , in meters per second .
6 . To calculate how high the ball went ( h ), take the vertical velocity squared and divide it by twice the gravitational acceleration : h = v y2
/ 2g
7 . You can also calculate the angle at which the ball left the ground ( θ ) by using a tiny
bit of trigonometry : θ = tan-1 ( v y v x
.)
RESULTS
You have calculated the speed at which you kicked the ball ( in meters / second ), the angle at which it launched , and how high you kicked it ( in meters ).
WHY ?
The above may sound like a bunch of mathematical calculations . But it ’ s based on a very simple , and very important , idea from physics : you can treat the vertical and horizontal motion of the ball independently . The total time spent in the air combined with how far along the ground the ball went tells you everything you need to know about the ball ’ s horizontal velocity .
SCIENCE PROJECT
27
Ignoring air resistance , the ball doesn ’ t experience any horizontal acceleration , so its horizontal velocity stays constant . The ball ’ s vertical motion is a different story . As soon it leaves your foot , gravity starts slowing the ball down . Eventually , the ball ’ s vertical velocity reaches zero . After that , the ball turns around and starts falling back to Earth , picking up speed the entire time . Ignoring air resistance ( again !), the ball ’ s vertical speed when it hits the ground is the same as its vertical speed when you kicked it .
Since the final and initial vertical speeds are the same , we can focus on just the second half of the ball ’ s trip . We can ask , how fast would a ball be moving after a certain amount of time if you dropped it from a great height ? That ’ s another way of asking how fast the ball is travelling when it hits the ground after falling from the highest part of its journey . The “ falling time ” is half the hang time . The total velocity comes from combining the horizontal and vertical velocities . We can draw the velocities like a right triangle . The horizontal and vertical velocities make up the sides of the triangle while the total velocity is its hypotenuse . Using the Pythagorean Theorem , you can use the sides to figure out the total speed with which the ball was launched . You can use the same triangle to figure out the angle at which it took off .
Figuring out the height comes back to just worrying about the ball ’ s vertical motion . We know how quickly the ball left your foot . And we know how strongly gravity is working to slow it down . That ’ s all we need to figure out how high the ball went . It ’ s the same as knowing how far your car will go if you ’ re driving at 60mph and suddenly hit the brakes . Except , in this case , the brakes are gravity !
To find height , velocity × time use kinemation equations
i ) v 2 = 2as ii ) v = u + at iii ) s = ut + 1
2 at2
Kinematic equation under gravity i ) v 2 – u 2 = ± 2gh ii ) v = u ± gt iii ) h = ut ± 1
2 gt2 ‘+’ → when object is travelling in the direction of gravity
‘–’ → when object is travelling in the opposite direction to the gravity
WINSPIRE : Empowering youth | January , 2017