A ball of mass 0.4 kg, initially at rest, is kicked directly toward a fence from a point 20 m away, as shown below.
The velocity of the ball as it leaves the kicker’s foot is 16 m/s at angle of 46◦ above the horizontal. The top of the fence is 4 m high. The ball hits nothing while in flight and air resistance is negligible.
The acceleration due to gravity is 9.8 m/s 2.
An object into the air at an angle to horizontal state two factor upon which the value of its range depends
To determine whether the ball clears the fence or not, we need to find the maximum height the ball reaches in its trajectory.
Step 1: Break down the initial velocity into horizontal and vertical components.
The initial velocity of the ball is given as 16 m/s at an angle of 46 degrees above the horizontal. We can use trigonometry to find the horizontal and vertical components of this velocity.
The horizontal component (Vx) can be found using the equation:
Vx = V * cosθ
where V is the magnitude of the velocity (16 m/s) and θ is the angle above the horizontal (46 degrees).
Vx = 16 * cos(46°)
Vx = 16 * 0.7193
Vx ≈ 11.509 m/s
The vertical component (Vy) can be found using the equation:
Vy = V * sinθ
where V is the magnitude of the velocity (16 m/s) and θ is the angle above the horizontal (46 degrees).
Vy = 16 * sin(46°)
Vy = 16 * 0.6947
Vy ≈ 11.115 m/s
Step 2: Determine the time it takes for the ball to reach its maximum height.
At the maximum height, the vertical velocity component (Vy) will be zero. We can use this information to find the time it takes for the ball to reach its maximum height.
Using the equation:
Vy = Voy + a * t
we can solve for t when Vy = 0.
0 = 11.115 m/s + (-9.8 m/s^2) * t
Rearranging the equation, we have:
9.8 t = 11.115
t ≈ 1.134 seconds
Step 3: Calculate the maximum height the ball reaches.
To find the maximum height, we can use the equation for vertical displacement:
Δy = Voy * t + (1/2) * a * t^2
Where:
Voy is the initial vertical velocity (11.115 m/s)
t is the time it takes to reach maximum height (1.134 seconds)
and a is the acceleration due to gravity (-9.8 m/s^2).
Δy = 11.115 m/s * 1.134 s + (1/2)(-9.8 m/s^2)(1.134 s)^2
Δy ≈ 6.166 meters
Step 4: Determine if the ball clears the fence.
The top of the fence is given as 4 meters high. If the maximum height the ball reaches (6.166 meters) is greater than the height of the fence (4 meters), then the ball clears the fence.
Therefore, the ball does clear the fence.