Question

Anthony is chosen to represent his school in the district level javelin competition. he throws the javelin at a distance of 3 meters from the ground. The height of the javelin is given by the function h(t)=-5t² 14t 3,where h is the height, in meters of the javelin and t is the time, in seconds.

Answer 1

The questions cover various aspects of projectile motion, including how different angles affect the range of a jumper, gravity's acceleration on another planet, and how to demonstrate that acceleration due to gravity is independent of an object's initial velocity.

Step-by-step explanation:

Projectile Motion and Acceleration due to Gravity

The student's questions pertain to projectile motion, specifically regarding the variation of the distance covered by a long jumper at different angles of projection, the calculation of acceleration due to gravity on a different planet, and the acceleration of an object in projectile motion. To find the distance lost by a long jumper taking off at 30° compared to 45°, we would use the projectile motion equations, considering that the maximum range for a projectile on Earth occurs at a 45° launch angle. Changing the launch angle to 30° would reduce the horizontal distance, which can be calculated. For the planet Arcon, with a maximum range given and initial launch speed, we would use the range formula to find the acceleration due to gravity. The acceleration of an object is independent of the object's velocity, so for an experiment, the student would need to measure the time it takes for each ball to hit the ground and compare these to show that they are the same despite different initial velocities.

Additionally, the motion of a shot putter's throw can be analyzed to find initial velocity and the net force during the throw, and the total distance a body travels under constant force can be expressed in terms of given variables. Understanding these principles enables us to solve problems related to projectile motion, such as those involving a javelin thrower like Anthony, whose javelin's height is determined by the quadratic function h(t) = -5t² + 14t + 3.

To graph a ball's velocities during projectile motion, a clear distinction between horizontal and vertical components must be made, with the horizontal velocity remaining constant and the vertical velocity being affected by gravity through time.

Answer 2

The javelin will reach its maximum height of 39 meters at t = 1.4 seconds.

Step-by-step explanation:

The height function of the javelin is given by h(t) = -5t² + 14t + 3. To find the maximum height, we need to determine the time (t) at which the height is maximized.

The maximum height occurs at the vertex of the parabolic function, and for a quadratic function in the form h(t) = at² + bt + c, the time at the vertex is given by t = -b/(2a).

In this case, the coefficient of t² is -5, and the coefficient of t is 14. So, the time at the vertex is given by t = -14/(2*(-5)) = 1.4 seconds. Substituting this time back into the original function gives us the maximum height: h(1.4) = -5(1.4)² + 14(1.4) + 3 = 39 meters.

Therefore, the javelin will reach its maximum height of 39 meters at t = 1.4 seconds.