A ball is thrown horizontally from the roof of a building 29.7 m high at 39.6 m/s. (a) What is the vertical velocity of the ball when it lands? (use down as the positive y-axis direction)b) What is the ball’s total velocity (speed and direction) when it lands?
To answer these questions, we can use a kinematic equation that relates the positions, velocities, and time for an object in motion. The equation we will be using is:
v_f = v_i + at
where:
v_f is the final velocity
v_i is the initial velocity
a is the acceleration
t is the time
Now let's break down the problem step by step:
a) What is the vertical velocity of the ball when it lands?
Given:
Initial vertical velocity (v_i) = 0 m/s (since the ball is thrown horizontally)
Final vertical position (y_f) = -29.7 m (negative since down is taken as the positive y-axis direction)
Time (t) = unknown
Vertical acceleration (a) = -9.8 m/s^2 (assuming no air resistance)
Using the kinematic equation, we can solve for the final velocity (v_f):
v_f = v_i + at
Since the initial vertical velocity is 0, the equation simplifies to:
v_f = at
Substituting the given values, we have:
-29.7 = -9.8t
Solving for t:
t = (-29.7) / (-9.8)
t = 3 seconds (approx)
Now we can find the final vertical velocity (v_f) by substituting the value of t in the equation:
v_f = at = (-9.8) * (3) = -29.4 m/s
So, the vertical velocity of the ball when it lands is approximately -29.4 m/s. The negative sign indicates that the ball is moving downward.
b) What is the ball's total velocity (speed and direction) when it lands?
To find the total velocity, we need to calculate both the vertical and horizontal velocities and then combine them using the Pythagorean theorem.
Since the ball is thrown horizontally, the horizontal velocity (v_horizontal) remains constant throughout the motion and is equal to the initial horizontal velocity.
Given:
Initial horizontal velocity (v_horizontal) = 39.6 m/s
To calculate the total velocity (v_total), we use the formula:
v_total = √(v_horizontal^2 + v_vertical^2)
Plugging in the values we've already found:
v_total = √((39.6)^2 + (-29.4)^2)
v_total = √(1563.36 + 864.36)
v_total = √(2427.72)
v_total ≈ 49.27 m/s
The total velocity (speed and direction) of the ball when it lands is approximately 49.27 m/s. The direction is determined by the angle at which the ball was thrown horizontally. Since the angle is not given in the problem, we can't determine the exact direction.