Calculate the impulse (in kgm/s) received by a football when it's kicked in a kickoff. The ball has a mass of 0.27 kg and travels a horizontal distance of 68 m. The ball is kicked at an angle q of 20.5 with the horizontal and spends 2.276 seconds in the air.
Vi is initial vertical velocity
u is horizontal velocity
S is initial speed
Do vertical problem first knowing it spends 2.276 s in air
That means it spends 1.138 seconds going up
v = Vi - g t
v = 0 at top
0 = Vi - 9.81 (1.138)
so
Vi = 11.16 meters/second
Is that enough?
remember tan 20.5 = Vi/u
so u = 29.8 meters/s
which you could also get from u = 68/2.276
S, initial speed = sqrt(u^2+v^2)
impulse = initial momentum
= .27 S
d = Xo*T = 68 m.
Xo * 2.276 = 68
Xo = 29.9 m/s. = Hor. component of
initial velocity.
Vo = Xo/cosA = 29.9/cos20.5 = 31.9 m/s =
Initial velocity.
Impulse = m*Vo = 0.27 * 31.9 = 8.61
kg.m/s.
To calculate the impulse received by the football, we first need to calculate the initial velocity of the ball. We can use the given information to find the initial horizontal and vertical velocities separately.
Given:
Mass of the football (m) = 0.27 kg
Horizontal distance traveled by the ball (d) = 68 m
Angle of projection (θ) = 20.5 degrees
Time spent in the air (t) = 2.276 s
Step 1: Calculate the initial horizontal velocity (v₀x).
The horizontal velocity remains constant throughout the motion. Therefore, we can use the formula: v₀x = d / t.
Substituting the values, we get:
v₀x = 68 m / 2.276 s = 29.85 m/s.
Step 2: Calculate the initial vertical velocity (v₀y).
The initial vertical velocity can be obtained using the formula: v₀y = v₀ * sin(θ), where v₀ is the initial speed.
To find the initial speed (v₀), we can use the formula: v₀ = d / t.
Substituting the values, we get:
v₀ = 68 m / 2.276 s = 29.85 m/s.
Now, substituting v₀ and θ into the formula, we get:
v₀y = 29.85 m/s * sin(20.5°) = 10.27 m/s.
Step 3: Calculate the total impulse (J).
Impulse (J) can be calculated using the formula: J = m * Δv, where m is the mass of the ball and Δv is the change in velocity.
Δv can be calculated with the formula: Δv = √(vₓ² + vᵧ²), where vₓ is the horizontal velocity and vᵧ is the vertical velocity.
Substituting the values, we get:
Δv = √(29.85 m/s)² + (10.27 m/s)² = √891.0225 + 105.4729 = √996.4954 = 31.57 m/s.
Finally, substituting the values into the impulse formula, we get:
J = 0.27 kg * 31.57 m/s = 8.52 kgm/s.
Therefore, the impulse received by the football when it is kicked in a kickoff is 8.52 kgm/s.