A person throws a rock straight up into the air. At the moment it leaves the person's hand it is going 88 mph. When the rock reaches its peak, how fast is it going and what is the magnitude and direction of its acceleration? Ignore air drag.
at it's peak the speed is zero, and the only acceleration is toward gravity, -9.81m/s^2
The velocity is zero at peak height, and the accleration is always -g (directed vertically downward).
To determine the speed of the rock at its peak and the magnitude and direction of its acceleration, we need to analyze the motion of the rock during its ascent.
Let's break down the problem step-by-step:
Step 1: Determine the acceleration due to gravity.
The acceleration due to gravity near the Earth's surface is approximately 9.8 m/s² downwards. Since we are given the value in mph, we need to convert it to meters per second. 1 mph is approximately equal to 0.447 m/s. Therefore, the initial speed of the rock is (88 mph) × (0.447 m/s per 1 mph) = 39.2 m/s upwards.
Step 2: Determine the maximum height reached by the rock.
At the peak of the toss, the rock momentarily comes to a stop before reversing its direction. At this point, its velocity is zero. The vertical displacement, or height reached, can be determined using the equation:
Δ𝑦 = 𝑣𝑖² / (2𝑔)
Where:
Δ𝑦 is the height reached,
𝑣𝑖 is the initial velocity (39.2 m/s),
and 𝑔 is the acceleration due to gravity (9.8 m/s²).
Plugging in these values, we have:
Δ𝑦 = (39.2 m/s)² / (2 × 9.8 m/s²) = 78.4 m
Therefore, the maximum height reached by the rock is 78.4 meters.
Step 3: Determine the magnitude and direction of acceleration at the peak.
Since the rock momentarily stops at its peak, its velocity is zero there. As it approaches the peak, its velocity decreases due to the gravitational pull, leading to a downward acceleration.
The magnitude of acceleration remains constant and is equal to the acceleration due to gravity (9.8 m/s²) throughout the rock's motion. The direction of acceleration at the peak is downwards.
Therefore, at the moment the rock reaches its peak, its speed is 0 m/s, and the magnitude of its acceleration is 9.8 m/s² downward.