To find the time when the ball is at a height of 4 meters, we can use the equations of motion under constant acceleration, taking into account the acceleration due to gravity. The value of acceleration due to gravity (g) is usually considered negative (-10 m/s²) when dealing with vertical motion because it acts in the downward direction.
Let's break down the problem and solve it step by step:
- Initial Velocity (u): The ball is thrown vertically upwards with a velocity of 12 m/s. Since the velocity is in the upward direction, we consider it positive.
u = 12 m/s
- Final Velocity (v): At the maximum height, the ball momentarily comes to rest before falling back down. The final velocity at this point is 0 m/s.
v = 0 m/s
- Acceleration (a): The acceleration due to gravity (g) acts in the downward direction. We consider it negative because it opposes the upward motion of the ball.
a = -10 m/s²
- Displacement (s): The ball reaches a height of 4 meters above its initial position. Since it is moving upwards, the displacement is positive.
s = 4 m
Now, let's use the equation of motion to find the time when the ball is at a height of 4 meters:
v = u + at
0 = 12 - 10t [Substituting the given values of v, u, and a]
10t = 12 [Rearranging the equation]
t = 12 / 10 [Dividing both sides by 10]
t = 1.2 seconds
Therefore, it takes approximately 1.2 seconds for the ball to reach a height of 4 meters.
Explanation of Negative Acceleration (Gravity):
Acceleration due to gravity (g) is considered negative when dealing with vertical motion because it acts in the opposite direction to the upward motion. This convention is chosen to maintain a consistent sign convention and mathematical framework.
When we consider upward motion, the positive direction is typically chosen as the direction of motion away from the ground. The force of gravity, acting in the downward direction, opposes this motion. Since acceleration is defined as the rate of change of velocity, and velocity is in the positive direction for upward motion, the acceleration due to gravity is given a negative sign to indicate its opposing effect.
By using negative acceleration, we can apply the equations of motion consistently. When the acceleration due to gravity is negative, it allows us to treat the upward direction as positive and the downward direction as negative, simplifying the calculations and maintaining the same mathematical framework for both upward and downward motion.
In summary, choosing the acceleration due to gravity as negative allows us to account for its opposing effect on upward motion and provides a consistent sign convention for vertical motion calculations.