Joshua throws a stone horizontally from a cliff at a speed of 20m/s, which strikes the ground 2.0 seconds later
a. use your knowledge of vectors and show that the stone strikes the ground at a speed of about 28m/s.
b. At what angles does the ball strike the ground
a. To find the speed at which the stone strikes the ground, we can use the equation for horizontal motion:
horizontal velocity = initial horizontal velocity = 20 m/s (since the stone is thrown horizontally)
The time taken for the stone to strike the ground is given as 2.0 seconds.
Using the equation for motion in the vertical direction:
vertical displacement = (initial vertical velocity) × time + (1/2) × (acceleration due to gravity) × (time)^2
Since the initial vertical velocity is 0 m/s (the stone is thrown horizontally), the equation becomes:
vertical displacement = (1/2) × (-9.8 m/s^2) × (2.0 s)^2
Simplifying this equation:
vertical displacement = -19.6 m
The negative sign indicates that the stone falls downward.
Now, we can find the final velocity of the stone when it strikes the ground using the equation for final velocity in the vertical direction:
final vertical velocity = (initial vertical velocity) + (acceleration due to gravity) × time
Since the initial vertical velocity is 0 m/s, the equation becomes:
final vertical velocity = (0 m/s) + (-9.8 m/s^2) × (2.0 s)
Simplifying this equation:
final vertical velocity = -19.6 m/s
The negative sign indicates that the final vertical velocity is downward.
Now, we can find the total speed (magnitude of the velocity) at which the stone strikes the ground using the Pythagorean theorem:
total speed = √(horizontal velocity)^2 + (final vertical velocity)^2
total speed = √(20 m/s)^2 + (-19.6 m/s)^2
Simplifying this equation:
total speed ≈ √(400 m^2/s^2 + 384.16 m^2/s^2)
total speed ≈ √784.16 m^2/s^2
total speed ≈ 28 m/s
Therefore, the stone strikes the ground at a speed of about 28 m/s.
b. Since the stone strikes the ground with a net downward vertical velocity, it means that it has an upward component of velocity as well.
To find the angle at which the stone strikes the ground, we can use the equation for the tangent of an angle:
tan(angle) = (final vertical velocity) / (horizontal velocity)
tan(angle) = (-19.6 m/s) / (20 m/s)
Simplifying this equation:
tan(angle) ≈ -0.98
To find the angle, we can take the inverse tangent (or arctan) of both sides of the equation:
angle ≈ arctan(-0.98)
Using a calculator:
angle ≈ -45 degrees or 135 degrees
Therefore, the stone strikes the ground at approximately -45 degrees (below the horizontal) or 135 degrees (above the horizontal).
To solve this problem, we can break it down into two parts:
a. Finding the horizontal and vertical components of the stone's initial velocity:
Since Joshua throws the stone horizontally, the initial velocity in the horizontal direction (Vx) is equal to 20 m/s.
In the vertical direction, the stone is subject to the acceleration due to gravity. We know that the time of flight is 2.0 seconds, so using the equation of motion for vertical motion:
d = Vit + 0.5gt^2
where d is the vertical distance (in this case, the height of the cliff) and g is the acceleration due to gravity (approximately 9.8 m/s^2), we can calculate the initial vertical velocity (Vy).
Assuming the height of the cliff is h, we have:
h = 0.5gt^2, since the initial vertical velocity is zero.
Solving for Vy:
h = 0.5 * 9.8 * (2.0)^2
h = 19.6 meters
Using the equation for vertical velocity:
Vy = gt
Vy = 9.8 * 2.0
Vy = 19.6 m/s
So the stone's initial velocity in the vertical direction (Vy) is 19.6 m/s.
b. Finding the speed and angle of impact:
To find the speed at which the stone strikes the ground, we can calculate the magnitude of the final velocity (Vf).
Using the kinematic equation for vertical motion:
Vf = Vy + gt
Vf = 19.6 + (9.8 * 2.0)
Vf = 19.6 + 19.6
Vf = 39.2 m/s
The speed at which the stone strikes the ground is approximately 39.2 m/s.
To find the angle at which the stone strikes the ground, we can use trigonometry. We know the horizontal component of the velocity (Vx) and the vertical component of the velocity (Vy). The angle can be calculated using the following formula:
θ = arctan(Vy / Vx)
θ = arctan(19.6 / 20)
θ ≈ 44.99 degrees
Therefore, the stone strikes the ground at an angle of approximately 44.99 degrees.