A football is kicked with an initial velocity of 25 m/s at an angle of 45 degrees with the horizontal . Determine the time of flight, the horizontal distance, and the peak height of the long jumper.
To determine the time of flight, horizontal distance, and peak height of the football, we can analyze its motion using the equations of motion.
Step 1: Resolve the initial velocity into horizontal and vertical components.
Given:
Initial velocity (V) = 25 m/s
Launch angle (θ) = 45 degrees
The horizontal component of the velocity (Vx) can be calculated using:
Vx = V * cos(θ)
Vx = 25 * cos(45)
Vx = 25 * √2 / 2
Vx = 25 * 0.707
Vx ≈ 17.68 m/s
The vertical component of the velocity (Vy) can be calculated using:
Vy = V * sin(θ)
Vy = 25 * sin(45)
Vy = 25 * √2 / 2
Vy = 25 * 0.707
Vy ≈ 17.68 m/s
Step 2: Determine the time of flight.
The time of flight (T) can be calculated using the formula:
T = (2 * Vy) / g
where g is the acceleration due to gravity (approximately 9.8 m/s²).
T = (2 * 17.68) / 9.8
T ≈ 3.61 seconds
Therefore, the time of flight of the football is approximately 3.61 seconds.
Step 3: Calculate the horizontal distance.
The horizontal distance (D) can be calculated using the formula:
D = Vx * T
D = 17.68 * 3.61
D ≈ 63.78 meters
Therefore, the horizontal distance covered by the football is approximately 63.78 meters.
Step 4: Determine the peak height.
The peak height (H) can be calculated using the formula:
H = (Vy²) / (2 * g)
H = (17.68²) / (2 * 9.8)
H = 312.87 / 19.6
H ≈ 15.97 meters
Therefore, the peak height reached by the football is approximately 15.97 meters.
To determine the time of flight, horizontal distance, and peak height of the football, we can use the equations of motion in projectile motion.
1. Time of flight:
The formula to calculate the time of flight in projectile motion is given by:
Time of flight = 2 * (initial vertical velocity) / (acceleration due to gravity)
In this case, the initial vertical velocity is given by:
Vertical component of initial velocity = initial velocity * sin(angle)
Given:
Initial velocity = 25 m/s
Angle = 45 degrees
First, convert the angle to radians by multiplying it by π/180:
Angle in radians = 45 * π / 180
Then, calculate the initial vertical velocity:
Vertical component of initial velocity = 25 m/s * sin(45 * π / 180)
Now, substitute the values into the time of flight formula:
Time of flight = 2 * (Vertical component of initial velocity) / (acceleration due to gravity)
2. Horizontal distance:
The horizontal distance covered by the football can be calculated using the formula:
Horizontal distance = (initial horizontal velocity) * Time of flight
In this case, the initial horizontal velocity is given by:
Horizontal component of initial velocity = initial velocity * cos(angle)
Now, substitute the values into the horizontal distance formula.
3. Peak height:
To calculate the peak height reached by the football, we can use the formula for vertical displacement in projectile motion:
Vertical displacement = (initial vertical velocity)^2 / (2 * acceleration due to gravity)
Now, substitute the value of the initial vertical velocity into the formula.
By following these steps, you will be able to calculate the time of flight, horizontal distance, and peak height of the football.
Good
Get the time of flight from the initial vertical velocity component. The times spent going up and down are equal.
Range is horizontal velocity component times time of flight.
Peak height is achieved when the vertical velocity component is zero.
Now go earn some honors.