I have two different questions I can't figure out.
1) Projectile motion: At what initial speed must the basketball player throw the ball at an angle of 55 degrees above the horizontal, to make the foul shot? The horizontal distance from the ball to the basket is 13 feet. Initial height of the ball is 7 ft and the final height which is the basket is 10 ft.
2)Force and Motion: A 12 kg armadillo runs onto a large pond of level, frictionless ice. The armadillo's initial velocity is 5m/s along the positive direction of the x-axis. Take its initial position on the ice as being the origin. It slips over the ice while being pushed by a wind with a force of 17N in the positive direction of the y-axis. In unit vector notation, what are the animal's (a) velocity and (b) position vector when it has slid for 3 seconds?
Assume 10 ft is the top of the trajectory. (Acually, a good free throw shooter would aim higher than that.) The ball must rise 3 feet in the time it takes to travel horizontally 13 feet. Let T be the travel time from release until it is at the basket. Let V be the release velocity.
Vcos55*T = 13 ft
Vsin55 = g T where g is the acceleration of gravity, 32.2 ft^2/s.
tan 55 = gT^2/13
T^2 = (13/32.2)tan55 = 0.577 sec^2
T = 0.759 sec
Now solve for V
1) To solve the projectile motion problem, we need to break it down into horizontal and vertical components.
a) Horizontal component:
The horizontal distance from the ball to the basket is given as 13 feet. We can use this information to find the time taken by the ball to reach the basket horizontally.
The horizontal motion does not experience any acceleration, so the horizontal velocity remains constant throughout the motion. We can use the equation:
Horizontal distance = horizontal velocity * time
Since the initial speed is the magnitude of the horizontal velocity, we can rewrite this equation as:
13 ft = initial speed * time
b) Vertical component:
To determine the initial speed (magnitude of the velocity), we need to analyze the vertical motion of the ball.
The initial height of the ball is given as 7 ft, and the final height (basket height) is given as 10 ft. The time taken for the ball to reach its peak height is the same as the time taken to fall from the peak to the final height.
We can use the equations of motion to calculate the time and initial speed in the vertical direction. The equations for vertical motion are:
Vertical displacement = initial vertical velocity * time + 0.5 * acceleration * time^2
In this case, the vertical displacement is the difference between the initial and final height, which is (10 ft - 7 ft) = 3 ft. The initial vertical velocity is given by:
initial vertical velocity = initial speed * sin(angle)
The acceleration is due to gravity and is approximately 32.2 ft/s^2.
By substituting the known values into the equation for vertical displacement, you can solve for the time taken.
Once you have found the time taken, you can substitute it back into the equation for horizontal distance to find the initial speed (magnitude of the velocity).
2) To solve the force and motion problem, we first need to determine the net force acting on the armadillo.
The force from the wind is given as 17N in the positive y-direction. Since the ice is frictionless, there are no other forces acting on the armadillo.
The net force can be calculated using vector addition:
Net force = force from wind
In unit vector notation, the force can be written as:
Net force = 0i + 17j
(a) Velocity vector:
To find the velocity vector, we need to combine the initial velocity with the net force. Since there is no acceleration in the horizontal direction, the velocity in the x-direction remains constant.
The initial velocity is given as 5 m/s along the positive x-axis, so the velocity vector can be written as:
Velocity vector = 5i + (force from wind/mass)*time
Since the force from the wind is given as 17N and the mass of the armadillo is given as 12 kg, we can substitute these values into the equation.
(b) Position vector:
To find the position vector after 3 seconds, we need to integrate the velocity vector with respect to time.
Position vector = initial position + integral of velocity vector over time
Given that the initial position is the origin (0, 0), and the initial velocity vector is 5i, we can integrate the velocity vector over time to find the position vector.
The position vector will be in the form:
Position vector = 5ti + (force from wind/mass)*(t^2)/2
Substitute the value of time (3 seconds) into the equation to calculate the position vector.