A ball is shot straight up from the top of a 15-story building. The motion of the ball could be described by the function: h(t) = -16t^2 + 144t + 160
where t represents the time the ball is in the air in seconds and h(t) represents the height in feet .
Why is the leading coefficient negative?
because gravity pulls downward.
The initial 144 ft/s gets reduced by the constant pull of gravity until at the vertex, the ball starts falling.
The leading coefficient (-16) in the given function, h(t) = -16t^2 + 144t + 160, is negative because it represents the acceleration due to gravity. In this context, the negative leading coefficient indicates that the ball is moving in the upward direction against the force of gravity.
The general equation to describe the vertical motion of an object under the influence of gravity is h(t) = -16t^2 + v0t + h0, where h(t) is the height of the object at time t, t represents the time in seconds, v0 is the initial velocity of the object, and h0 is the initial height.
In this particular case, since the ball is shot straight up from the top of a building, the initial velocity v0 is positive, representing the force applied to launch the ball upwards. However, as the ball reaches its highest point and starts coming back down, the force of gravity acts in the opposite direction, causing the ball to decelerate. This force of gravity is represented by the negative leading coefficient (-16) in the equation.