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Chapter 3: Problem 6

In a football game a kicker attempts a field goal. The ball remains in contact with the kicker's foot for \(0.050 \mathrm{~s}\), during which time it experiences an acceleration of \(340 \mathrm{~m} / \mathrm{s}^{2}\). The ball is launched at an angle of \(51^{\circ}\) above the ground. Determine the horizontal and vertical components of the launch velocity.

Short Answer

Expert verified
The horizontal component is approximately 10.71 m/s, and the vertical component is approximately 13.22 m/s.

Step by step solution

01

Determine final velocity

First, calculate the final velocity of the ball as it leaves the kicker's foot using the formula: \[ v = a \times t \]where:- \( v \) is the final velocity,- \( a = 340 \, \text{m/s}^2 \) is the acceleration,- \( t = 0.050 \, \text{s} \) is the time the ball is in contact with the foot.Substitute the values:\[ v = 340 \, \text{m/s}^2 \times 0.050 \, \text{s} = 17 \, \text{m/s} \]
02

Calculate horizontal component

Use the angle of launch to calculate the horizontal component of the velocity \( v_x \):\[ v_x = v \times \cos(\theta) \]where:- \( v = 17 \, \text{m/s} \) is the final velocity,- \( \theta = 51^\circ \) is the angle.Substitute the values:\[ v_x = 17 \, \text{m/s} \times \cos(51^\circ) \approx 10.71 \, \text{m/s} \]
03

Calculate vertical component

Use the angle of launch to calculate the vertical component of the velocity \( v_y \):\[ v_y = v \times \sin(\theta) \]where:- \( v = 17 \, \text{m/s} \) is the final velocity,- \( \theta = 51^\circ \) is the angle.Substitute the values:\[ v_y = 17 \, \text{m/s} \times \sin(51^\circ) \approx 13.22 \, \text{m/s} \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Kinematics
Kinematics deals with the motion of objects without considering the causes of this motion. In the context of projectile motion, understanding kinematics allows us to analyze the path of objects like footballs flying through the air.
This involves understanding quantities such as displacement, speed, velocity, and acceleration.
  • Displacement: The change in position of an object.
  • Velocity: How fast an object is moving and in what direction. For projectiles, initial velocity plays a crucial role.
  • Acceleration: The rate of change of velocity. In our exercise, acceleration helps us calculate the final velocity of the football as it leaves the foot.
By mastering kinematics, you can predict how long the ball stays in the air or how far it travels horizontally.
It sets the foundation for complex motion analysis in physics.
Horizontal and Vertical Components
When analyzing projectile motion, it's essential to break down the motion into horizontal and vertical components. This simplifies calculations since these two components are independent of each other.
  • Horizontal Component: This describes how far the object travels along the ground. Determined using the formula \( v_x = v \times \cos(\theta) \), it represents the consistent speed along the horizontal axis.
  • Vertical Component: This describes the object's climb and descent. Calculated using \( v_y = v \times \sin(\theta) \), it indicates the maximum height the ball reaches and the effect of gravity pulling it down.
These components allow us to map the path effectively. They help in understanding the actual trajectory and the impacts of gravity and velocity. By considering them separately, we can get precise results for the projectile's path.
Launch Velocity
Launch velocity is the initial speed at which the projectile is launched. It plays a critical role in determining how far and how high the projectile will travel.
  • Magnitude: Calculated using the formula \( v = a \times t \), where the ball's acceleration and contact time are considered.
  • Direction: The launch angle influences how the velocity is split into horizontal and vertical components. In this exercise, the launch angle is \(51^{\circ}\).
The combination of speed and direction affects the projectile's trajectory.
Understanding launch velocity helps in predicting the distance and height the projectile will achieve. By calculating the components separately, we ensure precise input for further kinematic equations and motion analysis.

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Most popular questions from this chapter

(a) A projectile is launched at a speed \(v_{0}\) and at an angle above the horizontal; its initial velocity components are \(v_{0_{x}}\) and \(v_{0_{y}}\). In the absence of air resistance, what is the speed of the projectile at the peak of its trajectory? (b) What is its speed just before it lands at the same vertical level from which it was launched? (c) Consider its speed at a point that is at a vertical level between that in (a) and (b). How does the speed at this point compare with the speeds identified in (a) and (b)? In each case, give your reasoning. A golfer hits a shot to a green that is elevated \(3.0 \mathrm{~m}\) above the point where the ball is struck. The ball leaves the club at a speed of \(14.0 \mathrm{~m} / \mathrm{s}\) at an angle of \(40.0^{\circ}\) above the horizontal. It rises to its maximum height and then falls down to the green. Ignoring air resistance, find the speed of the ball just before it lands. Check to see that your answer is consistent with your answer to part (c) of the Concept Questions.

Concept Simulation 3.2 at reviews the concepts that are important in this problem. A golfer imparts a speed of \(30.3 \mathrm{~m} / \mathrm{s}\) to a ball, and it travels the maximum possible distance before landing on the green. The tee and the green are at the same elevation. (a) How much time does the ball spend in the air? (b) What is the longest "hole in one" that the golfer can make, if the ball does not roll when it hits the green?

The escalator that leads down into a subway station has a length of \(30.0 \mathrm{~m}\) and a speed of \(1.8 \mathrm{~m} / \mathrm{s}\) relative to the ground. A student is coming out of the station by running in the wrong direction on this escalator. The local record time for this trick is 11 s. Relative to the escalator, what speed must the student exceed in order to beat the record?

Two cannons are mounted as shown in the drawing and rigged to fire simultaneously. They are used in a circus act in which two clowns serve as human cannonballs. The clowns are fired toward each other and collide at a height of \(1.00 \mathrm{~m}\) above the muzzles of the cannons. Clown A is launched at a \(75.0^{\circ}\) angle, with a speed of \(9.00 \mathrm{~m} / \mathrm{s}\). The horizontal separation between the clowns as they leave the cannons is \(6.00 \mathrm{~m} .\) Find the launch speed \(v_{0 B}\) and the launch angle \(\theta_{B}\left(>45.0^{\circ}\right)\) for clown B.

In diving to a depth of \(750 \mathrm{~m}\), an elephant seal also moves \(460 \mathrm{~m}\) due east of his starting point. What is the magnitude of the seal's displacement?
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