You might think it is easy to throw a ball straight up into the air. Actually, it’s
quite difficult to do this, more likely the ball moves a bit horizontally as well. To
describe this we need to talk about motion in 2 dimensions. Here is a simple
example: Imagine that a ball is tossed across the room along an arc. Neglecting
air resistance, where along the arc is the speed of the ball
a minimum? ________________
a maximum? ________________
Lab rooms usually have a level surface. Is that relevant for your answer? What –
if anything – would change if the room slopes downward at the far end?
Where along the path of the ball does the acceleration change? What is the
acceleration of the ball 0.78 seconds before it hits the ground?
To find the answers to these questions, we need to analyze the motion of the ball in two dimensions. Let's break it down step by step:
1. The speed of the ball is minimum at the highest point of its arc. This is because at the highest point, the ball momentarily stops moving vertically and starts its downward descent. However, the horizontal component of its velocity remains constant.
2. The speed of the ball is maximum at the lowest point of its arc. This is because at the lowest point, the ball has completed its downward descent and its vertical velocity is at its maximum. The horizontal component of its velocity remains constant throughout its motion.
3. The level of the room's surface is relevant for determining the shape of the ball's path but not for the speed of the ball. Regardless of the room's surface, neglecting air resistance, the speed of the ball will be affected only by its initial velocity and the acceleration due to gravity.
4. If the room slopes downward at the far end, it will not affect the answers to the first two questions, as the speed of the ball along the arc is independent of the slope of the room. However, the shape of the ball's path may change due to the influence of the slope.
5. The acceleration of the ball changes at every point along its path. The vertical component of acceleration remains constant throughout its motion and is equal to the acceleration due to gravity, which is approximately 9.8 m/s^2 downward. The horizontal component of acceleration is zero since there is no horizontal force acting on the ball if air resistance is neglected.
6. To determine the acceleration of the ball 0.78 seconds before it hits the ground, we need to use the equations of motion. The vertical component of velocity at that time can be found using the equation v = u + at, where v is the final velocity (which is zero as the ball hits the ground), u is the initial velocity, a is the acceleration, and t is the time. Since the ball is moving upward at the time, the initial velocity is positive. By substituting the given values, the equation becomes 0 = u + (-9.8 m/s^2) * 0.78 s.
By solving this equation, we can find the initial velocity of the ball.