The projectile motion of an object can be modeled using [tex]s(t)=g t^2+v_0 t+s_0[/tex], where [tex]g[/tex] is the acceleration due to gravity, [tex]t[/tex] is the time in seconds since launch, [tex]s(t)[/tex] is the height after [tex]t[/tex] seconds, [tex]v_0[/tex] is the initial velocity, and [tex]s_0[/tex] is the initial height. The acceleration due to gravity is -4.9 [tex]m / s ^2[/tex].
An object is launched at an initial velocity of 19.6 meters per second from an initial height of 24.5 meters. Which equation can be used to find the number of seconds it takes the object to hit the ground?
[tex]0=-4.9 t^2+19.6 t+24.5[/tex]
[tex]0=-4.9 t^2+24.5 t+19.6[/tex]
[tex]19.6=-4.9 t^2+24.5 t[/tex]
[tex]24.5=-4.9 t^2+19.6 t[/tex]
Answer (1)
Substitute the given values into the projectile motion equation.
Set the height s ( t ) to 0 to represent the object hitting the ground.
The resulting equation is 0 = − 4.9 t 2 + 19.6 t + 24.5 .
The equation to find the time it takes for the object to hit the ground is 0 = − 4.9 t 2 + 19.6 t + 24.5 .
Explanation
Understanding the Problem We are given the equation for projectile motion: s ( t ) = g t 2 + v 0 t + s 0 , where g = − 4.9 m / s 2 , v 0 = 19.6 m / s , and s 0 = 24.5 m . We want to find the equation that represents when the object hits the ground, which means s ( t ) = 0 .
Substituting the Values Substitute the given values into the equation: s ( t ) = − 4.9 t 2 + 19.6 t + 24.5 .
Setting s(t) to Zero Since we want to find the time when the object hits the ground, we set s ( t ) = 0 : 0 = − 4.9 t 2 + 19.6 t + 24.5 .
Final Equation The equation that can be used to find the number of seconds it takes the object to hit the ground is 0 = − 4.9 t 2 + 19.6 t + 24.5 .
Examples
Understanding projectile motion is crucial in fields like sports and engineering. For example, when designing a catapult or analyzing a baseball trajectory, engineers and athletes use similar equations to predict the range and height of projectiles. By adjusting initial velocities and launch angles, they can optimize performance, whether it's maximizing the distance of a thrown ball or ensuring a projectile lands accurately on target. This blend of physics and mathematics allows for precise control and prediction in real-world applications.
