A football player kicks a football with an initial velocity of 30m/s at an angle of 55 degrees relative to the flat ground.
a)what is the speed of the football at its highest point
b)find the highes point reached
c)find the total time in the air
d)how far is the football kicked (what is the net displacement)
e)what is the velocity of the ball when it lands on the ground?
c. 5s
To answer these questions, we can break down the motion of the football into its horizontal and vertical components. The horizontal component remains constant throughout the motion, while the vertical component is affected by gravity.
First, let's find the initial horizontal and vertical velocities of the football.
Given:
Initial velocity (vi) = 30 m/s
Angle (θ) = 55 degrees
a) Speed of the football at its highest point:
The highest point is reached when the vertical velocity becomes zero. At this point, the initial vertical velocity has been completely converted into the downward motion due to gravity.
First, let's find the initial vertical velocity (viy):
viy = vi * sin(θ)
viy = 30 m/s * sin(55 degrees)
viy ≈ 24.475 m/s
At the highest point, the vertical velocity is zero. Using the equation for vertical velocity (vfy) at the highest point:
vfy = viy + gt
0 = 24.475 m/s + (-9.8 m/s^2) * t
Solving for time (t):
t = 24.475 m/s / 9.8 m/s^2
t ≈ 2.5 seconds
Next, we find the speed at the highest point using the horizontal component:
Speed = horizontal component of velocity at the highest point
Speed = vi * cos(θ)
Speed = 30 m/s * cos(55 degrees)
Speed ≈ 15.43 m/s
So, the speed of the football at its highest point is approximately 15.43 m/s.
b) Highest point reached:
The highest point is reached at half of the total time taken, as the ball reaches its peak. Therefore, the highest point reached is given by:
Height = viy * t / 2
Height = 24.475 m/s * 2.5 seconds / 2
Height ≈ 30.59 meters
Thus, the highest point reached by the football is approximately 30.59 meters.
c) Total time in the air:
The total time in the air is twice the time it takes for the football to reach its highest point.
Total time = 2 * t
Total time = 2 * 2.5 seconds
Total time ≈ 5 seconds
So, the total time in the air is approximately 5 seconds.
d) Net displacement (horizontal distance traveled):
The horizontal distance traveled (net displacement) is given by the equation:
Displacement = horizontal component of velocity * total time
Displacement = vi * cos(θ) * total time
Displacement = 30 m/s * cos(55 degrees) * 5 seconds
Displacement ≈ 106.73 meters
Therefore, the net displacement of the football is approximately 106.73 meters.
e) Velocity of the ball when it lands on the ground:
The horizontal component of velocity remains constant throughout the motion. Therefore, the velocity of the ball when it lands on the ground is the same as the initial horizontal velocity.
Velocity = vi * cos(θ)
Velocity = 30 m/s * cos(55 degrees)
Velocity ≈ 15.43 m/s
Thus, the velocity of the ball when it lands on the ground is approximately 15.43 m/s.
To solve these problems, we can use the equations of motion for projectile motion. Let's go step by step:
a) To find the speed of the football at its highest point, we need to find the vertical component of the initial velocity. The vertical component can be found by multiplying the initial velocity (30 m/s) by the sine of the angle of projection (55 degrees). So, the vertical component is 30 * sin(55).
b) To find the highest point reached, we need to calculate the time it takes for the football to reach its highest point. The time of flight for an object launched with an initial velocity and angle can be calculated using the formula: time = 2 * vertical component of velocity / acceleration due to gravity. Since the ball reaches its highest point when the vertical component of velocity becomes zero, the total time to reach this point is half of the time of flight. Therefore, the highest point reached can be found using the formula: highest point = vertical component of initial velocity squared / (2 * acceleration due to gravity).
c) To find the total time in the air, we can find the time of flight, which is twice the time it takes for the football to reach the ground. The time of flight can be calculated using the formula: time = 2 * vertical component of initial velocity / acceleration due to gravity.
d) To find the net displacement, we need to calculate the horizontal component of the initial velocity. The horizontal component can be found by multiplying the initial velocity (30 m/s) by the cosine of the angle of projection (55 degrees). The net displacement is the horizontal component multiplied by the total time in the air.
e) Finally, to find the velocity of the ball when it lands on the ground, we can use the horizontal component of the initial velocity and the time of flight. The velocity of the ball when it lands is equal to the horizontal component of the initial velocity.
By plugging in the values into the formulas and performing the calculations, you can find the answers to each of the questions above.
You are not learning much when you have others find the formulas for you abd do the work. This is basic stuff. When you have tried some of these yourslef, we will be glad to help you.
For a), the vertical component is 0 and the horizontal component remains what it wa at kickoff. (Neglecting air friction)
c)Twice the time it takes to make the vertical velocity component zero.
d) (horizontal velocity component) x (time in the air)
For (e), why would the kinetic energy chamge? The height remains ground level.