4. A ball is kicked toward a fence from a point 32 meters away. The velocity of the ball as it leaves the kicker’s foot is 24 m/s at an angle of 37 degrees above the horizontal. The top of the fence is 2.0 meters high. Assuming air resistance to be negligible…(1 point)
a. What is the time for the ball to reach the plane of the fence?
b. What is the ball’s vertical position when it reaches the plane of the fence?
c. Draw graphs for vx vs t and vy vs t.
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To find the time for the ball to reach the plane of the fence, we can use the horizontal component of the velocity. The horizontal and vertical components are given by:
Vx = V * cos(θ)
Vy = V * sin(θ)
where V is the velocity magnitude and θ is the angle of the velocity vector.
In this case:
V = 24 m/s
θ = 37 degrees
Calculating the horizontal component of the velocity:
Vx = 24 m/s * cos(37 degrees) ≈ 19.12 m/s
Now, we can use the equation for horizontal motion:
X = X0 + Vx * t
where X is the position, X0 is the initial position, Vx is the velocity in the x-direction, and t is time.
Since the initial position (X0) is 32 meters, and the final position (X) is when the ball reaches the plane of the fence, we can write:
X = 0 meters (assuming the plane of the fence is at x = 0)
Setting up the equation:
0 = 32 meters + 19.12 m/s * t
Solving for t:
t = -32 meters / (19.12 m/s) ≈ -1.67 s
Since time cannot be negative, this means that the ball will take approximately 1.67 seconds to reach the plane of the fence.
Next, to find the ball's vertical position when it reaches the plane of the fence, we can use the equation for vertical motion:
Y = Y0 + Vy0 * t + 0.5 * a * t^2
Since the initial vertical position (Y0) is 0 meters, and the vertical velocity (Vy) at the start is given by:
Vy0 = V * sin(θ)
where V is the velocity magnitude and θ is the angle of the velocity vector, we can calculate:
Vy0 = 24 m/s * sin(37 degrees) ≈ 14.45 m/s
Since there is no vertical acceleration (assuming air resistance is negligible), we can simplify the equation to:
Y = Vy0 * t
Plugging in the known values:
Y = 14.45 m/s * 1.67 s ≈ 24.13 meters
So, the ball's vertical position when it reaches the plane of the fence is approximately 24.13 meters.
Finally, let's draw the graphs for Vx vs t and Vy vs t.
For Vx vs t, the horizontal component of the velocity (Vx) is constant, so the graph will be a horizontal line at the magnitude of Vx (in this case, approximate 19.12 m/s) at all points in time.
For Vy vs t, the vertical component of the velocity (Vy) changes due to the effect of gravity. It starts with an initial positive value, decreases until reaching zero at the maximum height, and then becomes negative. The graph will be a smooth curve that starts positive, reaches zero at the peak, and becomes negative.
It's important to note that the scales and units should be appropriately labeled on each axis of the graphs.