An object is thrown upward at time t=0 . After t seconds,...

An object is thrown upward at time $t=0 .$ After $t$ seconds, its height is $y=-4.9 t^{2}+7 t+1.5$ meters above the ground.
(a) From what height was the object thrown?
(b) What is the initial velocity of the object?
(c) What is the acceleration due to gravity?

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Key Concepts

Projectile Motion

Projectile motion refers to the motion of an object under the influence of gravity, where its path is described by a quadratic equation in time. This concept connects the initial position, initial velocity, and constant acceleration, typically resulting from gravitational force, which collectively determine the object's trajectory.

Initial Conditions in Quadratic Equations

In equations describing projectile motion, the constant term represents the initial height from which the object is thrown, while the linear coefficient represents the initial velocity. Evaluating the quadratic function at time zero provides the starting position, and the coefficient of the first degree term gives the initial rate of change of the object's position.

Acceleration Due to Gravity

In projectile equations, the quadratic term’s coefficient is linked to the acceleration acting on the object. Specifically, the quadratic coefficient is half the acceleration, so identifying it allows one to determine the acceleration due to gravity. This provides insight into the force acting in the vertical direction, typically resulting in a constant downward acceleration.

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