A soccer ball is kicked from the ground with an initial speed of 14.5 m/s. After 0.272 seconds, ts speed is 12.4 m/s. (A) Give a strategy that will allow you to calculate the ball's initial direction of motion. (B) Use your strategy to find the initial direction.
v₀⒳= v₀•cosα,
v₀⒴ =v₀•sinα,
v₀² =v₀⒳² +v₀⒴²,
v² =v⒳² +v⒴²,
v⒳= v₀⒳,
v⒴=v₀⒴ -g•t,
v² =v⒳² +v⒴² = v₀⒳²+(v₀⒴ -g•t)²=
=v₀⒳²+v₀⒴² -2 v₀⒴•g•t +(g•t)² =
= v₀²-2 v₀•sinα•g•t +(g•t)².
sinα= [v₀²-v²+(g•t)²]/2•v₀•g•t =
=[14.5² -12.4²+(9.8•0.272²]/(2•14.5•9.8•0.272) =
=0.823.
α=55.35°
To determine the initial direction of motion of the soccer ball, we can use the concept of projectile motion. Projectile motion involves two independent components: horizontal motion (along the x-axis) and vertical motion (along the y-axis).
(A) Here is a strategy to calculate the ball's initial direction of motion:
1. First, identify the independent components of motion. In this case, the horizontal motion (along the x-axis) represents the initial direction of the soccer ball.
2. Next, calculate the horizontal velocity at t = 0.272 seconds using the principle that horizontal velocity remains constant throughout projectile motion. Since there is no acceleration in the horizontal direction, the initial and final velocities will be the same. Therefore, the horizontal velocity at t = 0.272 seconds will be equal to the initial horizontal velocity of the ball.
3. Finally, using the calculated horizontal velocity, we can determine the initial direction of motion of the soccer ball by considering the positive or negative sign.
(B) Let's use the given information to find the initial direction of the ball:
Given:
Initial speed of the ball (v0) = 14.5 m/s
Speed of the ball after 0.272 seconds (v) = 12.4 m/s
1. Calculate the horizontal velocity at t = 0.272 seconds:
Since the horizontal velocity remains constant, the horizontal velocity at t = 0.272 seconds is equal to the initial horizontal velocity of the ball.
Horizontal velocity (vx) = Initial horizontal velocity = 14.5 m/s
2. Determine the initial direction of motion:
If the horizontal velocity (vx) is positive, the initial direction of motion is to the right.
If the horizontal velocity (vx) is negative, the initial direction of motion is to the left.
In this case, since the horizontal velocity is positive (14.5 m/s), the initial direction of the soccer ball is to the right.