a kicker kicks a football from the 5 yard line to the 45 yd line ( on the same half of the field). Ignoring air resistance, where along the trajectory is the speed of the football a minimum. what formuls is used and show work
a) at the 5 yd line, just after the football leaves the kicker's foot
b)at the 45 yd line, just before the football hits the ground
c)at the 35 yd line when the ball is coming down
d) at the 25 yd line , when the ball is at the top of its trajectory
What is the correct answer?
To determine where along the trajectory the speed of the football is a minimum, we need to consider the physics of projectile motion. The speed of an object at any point in its trajectory can be calculated using the formula:
v = √(v0^2 + 2ad)
Where:
- v is the final speed
- v0 is the initial speed
- a is the acceleration
- d is the displacement
In this case, we have two points along the trajectory: the 5 yard line (where the ball is kicked) and the 45 yard line (where the ball is about to hit the ground).
Let's calculate the speed at each of these points and determine which is a minimum.
a) At the 5 yard line, just after the football leaves the kicker's foot:
- Assuming the initial speed (v0) is constant, we can calculate the acceleration (a) using the distance (d) and the initial speed (v0).
- Since we are ignoring air resistance, there is no significant force acting on the ball horizontally, so the acceleration is zero.
- Using this information, we substitute the values into the formula: v = √(v0^2 + 2ad).
- With a = 0, the formula reduces to: v = √v0^2.
- Therefore, the speed at the 5 yard line is v0.
b) At the 45 yard line, just before the football hits the ground:
- Now we consider the displacement (d) as the distance from the 5 yard line to the 45 yard line.
- Again, we assume the initial speed (v0) is constant.
- The acceleration in the vertical direction is due to gravity, which is approximately -9.8 m/s^2 or -32 ft/s^2.
- Substituting these values into the formula: v = √(v0^2 + 2ad).
- Since a is negative (due to gravity), the acceleration term will result in a decrease in speed.
- Therefore, the speed at the 45 yard line is the minimum.
Hence, the answer is:
b) At the 45 yd line, just before the football hits the ground.